(CNN) -- A sharp exchange over the U.S. response to the September 11 terrorist attack in Libya dominated the political discussion on Wednesday following a bruising debate that analysts and polls scored a victory for President Barack Obama over Republican challenger Mitt Romney.
The Tuesday night debate at Hofstra University in Hempstead, New York, featured a revitalized Obama fighting back against his equally combative GOP foe in an argumentative encounter three weeks before Election Day.
With a third and final debate next week, the candidates appeared likely to secure their standing in an already tight race that portends a cliffhanger presidential vote.
On Wednesday, both campaigns continued their focus on battleground states considered crucial to winning the White House.
Obama headed to Iowa, where he took aim at Romney proposals at an event at Cornell College in Mount Vernon.
"His tax plan doesn't add up. His jobs plan doesn't create jobs. His deficit reduction plan adds to the deficit," Obama said. "So Iowa -- everybody here has heard of the New Deal, you've heard of the Fair Deal, you've heard of the square deal - Mitt Romney is trying to sell you a sketchy deal."
Romney campaigned in Virginia, where he said the president was out of ideas after what he called four years of failed leadership.
"I think it is pretty clear that when it comes to his policies and his answers and his agenda he is pretty much running on fumes," Romney told a campaign event in Chesapeake.
His running mate, conservative Rep. Paul Ryan, offered a similar attack line at his own event in Ohio, saying sluggish growth and high unemployment "may be the best President Obama can give us, but it's not the best we can give ourselves."
Obama also will campaign in Ohio later Wednesday and Vice President Joe Biden heads to Colorado and Nevada.
In the second debate, Obama rebounded from a lackluster performance in the initial meeting with Romney two weeks ago in Denver.
Obama forcefully defended his policies and challenged Romney on shifting positions on key issues while arguing his Republican rival's proposals would favor the wealthy if elected on November 6.
Romney repeatedly attacked Obama's record, saying millions of unemployed people and a sluggish economic recovery showed the president's policies had failed.
On the sensitive topic of the Libya attack that killed Ambassador Christopher Stevens and three other Americans, Romney suggested the Obama administration played politics by failing to immediately acknowledge what happened.
Obama shot back that the suggestion anyone in his administration would play politics on such an issue was "offensive." When Obama said he called it an act of terror shortly after it occurred, Romney challenged him, and Obama responded by saying "check the transcript."
Moderator Candy Crowley, CNN's chief political correspondent, cut in to say both men were right -- Obama referred to an act of terror shortly after the attack, but the administration took longer to fully explain what occurred.
Ryan challenged that assessment on Wednesday, telling CBS that Obama's initial remarks on the Benghazi attack were "a passing reference to acts of terror in general."
He noted that Obama and others, including U.N. Ambassador Susan Rice, subsequently suggested the violence spawned from a protest over an anti-Islam video before clarifying two weeks later it was a terrorist attack.
"The facts just don't square with that line of argument," Ryan said.
Another Republican, Rep. Phil Gingrey of Georgia, told CNN that Crowley's intervention helped Obama.
"That may have thrown Governor Romney off his game just for a second," Gingrey said Wednesday, insisting later that whether intended or not, Crowley "did aid and abet President Obama in that exchange."
Biden and a top Obama campaign strategist called the Republican criticism an attempt to play politics with a tragic issue.
"It became so clear to the American people how Governor Romney and the campaign continue to try to politicize a tragedy," Biden told ABC. "And their strategy seems to be to make it appear that the president didn't care, didn't know, or was lying. The fact of the matter is, the president was clear. We are going to get to the bottom of this. The whole world will know it."
Robert Gibbs, a former White House press secretary now helping the Obama campaign, said Romney's response to the Libya attack and other anti-U.S. violence in Egypt included an initial erroneous statement and now what he called trying "like Olympic gymnasts to politicize the issue by doing whatever they have to do."
Romney "has handled this thing enormously poorly from his very first response all the way through this debate," Gibbs told CNN, adding that the former Massachusetts governor "doesn't look look a strong commander in chief."
The Romney campaign seeks to make the Libya attack a major campaign issue to try to cut into Obama's advantage over Romney on foreign policy issues, according to polls.
The third and final presidential debate next Monday in Florida will focus on foreign policy.
Both candidates walked the floor with microphones in hand during the 90-plus minute debate on Tuesday, raising their voices at times and repeatedly challenging each other's points.
Crowley tried in vain at times to prevent each man from going over allotted time, with Obama speaking for more than three minutes longer than Romney overall. However, a count by CNN showed that for the second straight debate, Romney spoke more words despite talking for a shorter time.
Obama was on the attack from the start, but waited until his final answer -- with no chance for Romney to respond -- to raise his opponent's controversial "47%" comments at a fundraiser in May.
In remarks made public by a secretly recorded video of the event, Romney described 47% of the country as people dependent on government aid who refused to take personal responsibility. The president was criticized after the first debate for not raising the issue and he made sure to do so this time.
"Think about who he was talking about," Obama said, listing people on Social Security "who've worked all their lives," veterans "who've sacrificed for this country," students, soldiers and "people working hard every day."
The president said he wanted to fight for those people "because if they succeed, I believe the country succeeds."
Earlier, Obama went after Romney's five-point economic plan that the GOP candidate repeated two times during the debate, saying it really was a one-point plan "and that plan is to make sure that folks at the top play by a different set of rules."
Romney shot back that "if you elect President Obama, you know what you're going to get -- you're going to get a repeat of the last four years."
Over and over, Romney described what he called the failings of Obama's policies including rising federal deficits and debts, more than 20 million people unemployed and anemic economic growth.
"We don't have to settle for what we're going through," Romney said at one point. "We don't have to settle for gasoline at four bucks. We don't have to settle for unemployment at a chronically high level. We don't have to settle for 47 million people on food stamps. We don't have to settle for 50 percent of kids coming out of college not able to get work. We don't have to settle for 23 million people struggling to find a good job."
Obama was "great as a speaker, but his policies don't work," Romney said. Attempting to rebut Obama's criticism of his own policies, Romney insisted he would prioritize middle class growth, saying "it's about how we can get the middle class of this country a bright and prosperous future."
However, Romney failed to provide further specifics of his tax policy, even when one audience member asked about unspecified deductions and loopholes the candidate says he will eliminate.
A CNN/ORC International poll of people who watched the debate indicated that 46% thought Obama won, compared to 39% for Romney. The result was within the survey's margin of error, and responses to other questions showed debate watchers favored Romney on the economy and other major issues.
After the first debate on October 3, a similar poll showed Romney scored a solid victory in the eyes of more than 60% of respondents.
"Most improved -- that award goes to Barack Obama," CNN Senior Political Analyst David Gergen said.
Daily Beast's Andrew Sullivan, who called Obama's poll numbers after the first debate "devastating," predicted the president would come "kicking back in the polls" in coming days.
Erick Erickson, the conservative RedState.com blogger and CNN contributor, thought Romney won the debate based on "clear majorities outside the margin of error" in the CNN/ORC poll who thought Romney would be better for the country on economic issues.
"In fact, while other areas of the debate may overshadow this point, Romney deftly dispatched Obama on his economic record," Erickson said, calling it "the one issue that matters."
An awkward phrase by Romney in addressing gender pay inequality was creating the most buzz around the debate.
Romney said when he was elected governor of Massachusetts, all the applicants for cabinet positions were men, so he sought out women applicants. "I went to a number of women's groups and said, 'Can you help us find folks?' and they brought us whole binders full of women."
Before the debate was over "binders full of women" had a Twitter hashtag, a series of memes on Tumblr, and a Facebook page with over more than 100,000 fans. The phrase was the third-fastest rising search on Google during the debate.
Unlike the first presidential debate, the town hall-style format allowed audience members to ask the questions. Crowley, the first woman to serve as moderator of a presidential debate in 20 years, tried to get in as many of the questions as possible from the uncommitted voters in the hall, sometimes struggling to cut off the candidates as they tried to make points or argued with each other.
Questions addressed troubles finding jobs, tax policy and immigration, with Romney saying Obama failed to deliver on a promise to pass a major immigration overhaul while the president said Romney backed conservative positions such as opposing a measure that gives some children of unregistered immigrants a path to legal status.
Obama repeated his past criticism that Romney's tax plan doesn't add up, saying "when he's asked how are you going to do it, which deductions, which loopholes are you going to close, he can't tell you."
"You're going to be paying for it," Obama said of middle class taxpayers. "You're going to lose some deductions, and you can't buy the sales pitch. Nobody who's looked at it that's serious actually believes it adds up."
Romney, meanwhile, repeatedly insisted his plan won't raise taxes for middle class Americans but also won't raise or lower the tax burden on the wealthy, in contrast to Obama's proposal to let tax rates on the top 2% of wage earners return to higher tax rates from the Clinton era.
The president needed a strong debate to try to blunt Romney's rise in the polls after their first showdown. The most recent CNN "poll of polls" -- an aggregate of the latest major surveys -- showed Romney with a slight edge nationally at 48%-47%.
In the battleground states considered up-for-grabs, polls show Romney has narrowed Obama's lead or caught the president.
CNN's Keating Holland and Kevin Liptak contributed to this report.