Washington (CNN) -- Senate Majority Leader Harry Reid called on Republicans to cooperate with an expanded Democratic majority Wednesday, a day after the Democrats picked up at least one more seat in the chamber.
"It is better to dance than to fight," said Reid, D-Nevada. "It is better to work together. Everything doesn't have to be a fight."
The last votes were still being counted Wednesday in some Western states. CNN now projects Democrat Heidi Heitkamp will beat Republican Rick Berg in North Dakota's Senate race, keeping that seat in Democratic hands after the retirement of longtime Sen. Kent Conrad.
In addition, Montana Sen. Jon Tester was projected to hold off a challenge from Republican congressman Denny Rehberg to win a second term there.
Reid spoke as the outgoing Congress is under pressure to head off a looming combination of tax increases and across-the-board cuts to federal agencies slated to kick in the first week of January. His deputy, Sen. Dick Durbin, said Tuesday's results are hugely important when it comes to that "fiscal cliff."
"We need a more positive outlook to solve our problems," said Durbin, of Illinois. "A stalemate is impossible."
Republicans captured a seat held by retiring Democrat Ben Nelson in Nebraska, where tea party-backed Republican Deb Fischer beat Bob Kerrey, a former Democratic governor and senator.
But Democrats will be replacing Republicans in Massaschusetts, where Sen. Scott Brown lost his bid for a full term to former White House adviser and bailout watchdog Elizabeth Warren; and in Indiana, where GOP veteran Richard Lugar lost a Republican primary to Richard Mourdock, who was beaten by Democratic Rep. Joe Donnelly.
The national spotlight shone on Mourdock in the waning weeks of the campaign when he discussed his views on abortion during a debate. He said: "I've struggled with it myself for a long time, but I came to realize that life is that gift from God. And even when life begins in that horrible situation of rape, that it is something that God intended to happen."
"I was attacked for standing on my principles," he said.
A Democrat also replaces one of the Senate's two independents, both of whom align themselves with the Democratic caucus, when Connecticut's Pat Murphy takes over the seat now held by Sen. Joe Lieberman.
That leaves a Senate with 53 Democrats and one independent, Vermont's Bernie Sanders -- up from the 51 Democrats and two independents in the current chamber. And voters in Maine sent another independent to Washington on Tuesday, with former Gov. Angus King taking the place of retiring Republican Sen. Olympia Snowe.
King has studiously refused to say which caucus he'll join. Though both parties say they expect him to align with the Democrats, he says he'll meet with Reid and with Senate Minority Leader Mitch McConnell, R-Kentucky, before making a decision.
"I hope I can be a bit of a bridge between the two parties," King told CNN. "When I was governor, I was an independent and worked with both sides and was able to achieve quite a bit. There were times when I agreed more with the Democrats and times I agreed more with the Republicans."
Democrats also held onto seats in Virginia and Ohio and saw several firsts. Rep. Tammy Baldwin's victory in Wisconsin will make her the first openly gay senator, in addition to being the state's first female senator. Hawaii also elected its first female senator when Democratic Rep. Mazie Hirono won, CNN projects.
There will be a record number of women in the new Senate class -- at least 19, based on CNN projections and current members.
Two years ago, Republicans had every reason to believe they could take back the Senate this year, after major midterm election gains. Going into Tuesday's vote, Republicans were protecting only 10 seats, while Democrats were defending 23 -- many in narrowly divided swing states.
But Republicans fell short of winning the seats they would need to tip the balance of power in the Senate, as controversy clouded the final days of campaigning in several races.
In August, the campaign of Rep. Todd Akin nearly collapsed after the Missouri Republican's comments about "legitimate rape" and his suggestion that women could biologically prevent pregnancy if they are raped. Until then, Republicans believed Akin would defeat Democratic Sen. Claire McCaskill, who polls showed was not very popular after just one term. McCaskill defeated Akin at the polls Tuesday, CNN projected.
Key Senate race snapshots
Arizona: Rep. Jeff Flake (R) vs. Richard Carmona (D)
Final result: Rep. Jeff Flake wins
Flake, a vocal fiscal conservative, will take the seat now held by retiring GOP Sen. Jon Kyl.
Carmona, a Hispanic Vietnam veteran and surgeon general under President George W. Bush, proved to be a formidable opponent for Flake, a six-term congressman who had to fight his way through a tough August primary. Tight general election polls caused both campaigns to go negative. Among other things, Flake accused Carmona of having anger management problems, while Carmona argued Flake failed to adequately support veterans while in the House.
No Democrat has won a U.S. Senate seat in Republican-leaning Arizona since 1988.
Connecticut: Rep. Chris Murphy (D) vs. Linda McMahon (R)
Final result: Rep. Chris Murphy wins
In the closely watched senate race in Connecticut, Democratic Rep. Chris Murphy defeated Republican nominee and professional wrestling executive Linda McMahon, according to a CNN projection.
Murphy will take over the Senate seat being vacated by retiring Sen. Joe Lieberman, an Independent who sided with Democrats.
His win preserves a seat for Democrats who feared losing it to the wealthy McMahon, who spent over $90 million in the last two elections to try to win a Senate seat.
The race was marked by heavy negative advertising.
Indiana: Richard Mourdock (R) vs. Rep. Joe Donnelly (D)
Final result: Rep. Joe Donnelly wins
Democrats scored a major upset in the Indiana Senate race, scooping up a seat in the deeply red state Republicans had expected to hold.
The campaign appeared to turn for the Democrats after the GOP candidate made controversial comments about rape during a televised debate.
Mourdock, the state treasurer, was backed by both tea party supporters and the Republican establishment when he defeated in the primary longtime GOP moderate Sen. Richard Lugar. When that happened, Democrats thought Donnelly, who is conservative and pro-life, had an outside chance of winning the seat.
Mourdock gave Donnelly an opening when during a debate he defended his opposition to abortion even in the case of rape because, "I think even when life begins in that horrible situation of rape, that is something God intended to happen." Top Democrats and Republicans criticized the remark and within days, Donnelly took a substantial lead in one respected Indiana poll.
Maine: Charlie Summers (R) vs. Cynthia Dill (D) vs. former Gov. Angus King (I)
Final result: Former Gov. Angus King wins
Former Gov. Angus King, an Independent, steadfastly has refused to say if he will organize with Democrats or Republicans when he gets to Washington.
King will replace Sen. Olympia Snowe, herself an independent-minded Republican who stunned her colleagues this year when she announced she would retire from the Senate, which she complained has become increasingly partisan.
Democrats are convinced King will side with them. As evidence, they point to King's support for President Barack Obama and many of his policies. In a sign of that confidence, national Democrats ignored the Democratic candidate Dill and spent heavily against Summers in order to help King.
Republicans also believe that King will caucus with Democrats, so they spent millions to defeat him.
But an aide told CNN last week that King still has not made up his mind and won't until he gets to Washington to see the state of play.
Massachusetts: Sen. Scott Brown (R) vs. Elizabeth Warren (D)
Final result: Elizabeth Warren wins
In a key pick-up of a Republican Senate seat, Warren defeated Brown Tuesday, according to a CNN projection.
Warren, a Harvard University law professor and former Obama administration financial consumer advocate, beat the freshman senator who was seeking a full term to the seat he won two years after the death of Democratic senator Ted Kennedy. Brown, the first GOP Senator from deep blue Massachusetts in decades, billed himself as an independent who voted for the needs of Massachusetts, not the demands of either party.
The Warren-Brown race was one of the costliest and nastiest campaigns this cycle. It was marked by Brown's repeated accusations that Warren claimed to be a minority in order to advance her career. She denied that was her motive for listing her ancestry as Native American, saying she was told about her heritage by her family.
Montana: Sen. Jon Tester (D) vs. Rep. Dennis Rehberg (R)
Final result: CNN projects Tester wins
Democrat Jon Tester knew he'd have an uphill battle to a second term against six-term Rep. Denny Rehberg. Both are known quantities with high favorables in the state. Both campaigns went negative in the fight over Social Security, and both said the other is distorting their views.
Nebraska: Former Sen. Bob Kerrey (D) vs. Deb Fischer (R)
Final result: Deb Fischer wins
Republicans argued that Kerrey, once a popular governor and two-term senator, lost touch with the conservative state after spending years living in liberal New York City, where he worked as president of the New School. The conservative super PAC American Crossroads began running ads against Kerrey even before he launched his campaign.
For his part, Kerrey tried to overcome the state's GOP tilt in a presidential election year by stressing his commitment to entitlement reform, a balanced budget, and other conservative fiscal issues. He also won a surprise late endorsement from former Nebraska Republican Sen. Chuck Hagel, considered by many to be a GOP renegade.
Republicans began eyeing Nelson's seat even before he announced his retirement. Fischer surprised GOP leaders by squeezing out a primary win over two opponents considered stronger at the start of the year.
Nevada: Sen. Dean Heller (R) vs. Rep. Shelley Berkley (D)
Final result: Dean Heller wins
Democrats' strength in Nevada was put to the test once again in this Senate election. With voter registration numbers on their side, seven-term Democratic Rep. Shelley Berkley had hoped to deny Republican Sen. Dean Heller a full Senate term (Heller was appointed to the seat after Republican John Ensign resigned).
A House ethics investigation and a lackluster debate performance kept Berkley below Heller in recent polls. Heller's strong fund-raising ability also helped him keep the advantage in a state with a heavy Latino population that normally overwhelmingly votes Democratic.
North Dakota: Heidi Heitkamp (D) vs. Rep. Rick Berg (R)
CNN projects Heitkamp wins
Former state Attorney General Heidi Heitkamp gave Democrats their best opportunity to keep the seat of retiring five-term Sen. Kent Conrad. Republicans hoped first-term Rep. Rick Berg will repeat the party's success of 2010, when they took over retiring Democrat Byron Dorgan's seat.
The state with the lowest unemployment in the country has a history of split-ticket voting, giving Heitkamp an opportunity to show her independence from national Democrats. She's publicly disagreed with Obama on issues like energy, which she points to as evidence she won't be a rubber stamp. She enjoyed high favorables and solid support from Republican ticket-splitters and has hammered Berg for his connection to a controversial real estate company.
Ohio: Sen. Sherrod Brown (D) vs. Josh Mandel (R)
Final result: Sen. Sherrod Brown wins
Primarily because of those frequent trips from President Barack Obama and Republican candidate Mitt Romney, the contest between Brown and Mandel played out in Ohio against the backdrop of the presidential election.
Fueled by Super PAC money and advertising, the race got nasty at times. In total, $67 million dollars was spent in the election, according to the Center for Responsive Politics. Much of that -- $34 million -- was sent by outside groups advocacy groups.
At the three debate, Mandel and Brown battled over the auto bailout, abortion and each other's record. The testiness of the race spilled out on the debate stage -- at one debate Mandel told Brown to "calm down" and called him "a liar."
Brown was first elected to the Senate in 2006, during a wave of Democratic victories in the Senate and House. Throughout most of the 2012 race, especially in September and October, Brown led Mandel by around 10 percentage points in independent poling.
Pennsylvania: Sen. Bob Casey (D) vs. Tom Smith (R)
Final result: Sen. Bob Casey wins
Though recent polls gave Republicans hope where they previously had little, freshman Democratic Sen. Bob Casey of Pennsylvania was re-elected Tuesday.
GOP candidate Tom Smith, a wealthy business executive, lagged far behind Casey for much of the race.
In the last few weeks, he spent heavily on advertising and was able to narrow the polls. But he never took the lead
Virginia: Former Gov. Tim Kaine (D) vs. former Gov./Sen. George Allen (R)
Final result: Gov. Tim Kaine wins
Allen conceded in a Tuesday night speech and pledged his support to Kaine.
"I congratulated him and pledged my support and cooperation as he undertakes the solemn vows of representing the people of Virginia during very difficult times in our nation's history," Allen said. "We haven't succeeded my friends in winning this election."
The race pitted two former Virginia governors against one another and each pulled from their experience in the state house throughout the race.
Though, Kaine linked himself to Obama and the president's campaigning in Virginia, he also painted himself as someone who had worked with both Democrats and Republicans. Allen, however, tried to turn that link against Kaine -- tying him to Obama and the Democratic Party, where the former Democratic governor served as chairman.
Making that case was costly. When outside and candidate spending was added together, the Kaine-Allen race was the most expensive race in the country. Outside groups, on their own totaled over $50 million, according to the Center for Responsive Politics.
Wisconsin: Rep. Tammy Baldwin (D) vs. former Gov. Tommy Thompson (R)
Final result: Rep. Tammy Baldwin wins
Baldwin, along with outside groups, spent a lot of advertising dollars painting Thompson as someone who left politics to cash in on lobbying. Both Democrats and Republicans officials acknowledge that the attacks drove Thompson's negatives up -- and led to a testy campaign tone.
The Wisconsin Senate seat opened up when Sen. Herb Kohl retired after four terms in the Senate. Thompson, who won a four-way primary for the Republican nomination, was Baldwin's first competition in the race -- she ran unopposed for the Democratic nomination.
The election win makes Baldwin the first openly gay member of the Senate.
CNN's Alan Silverleib, Dana Bash, Dan Merica, Adam Levy and Robert Yoon contributed to this report.