"This is no longer just in our dreams," one man shouted.
And the leader of the military-backed ruling party effectively conceded defeat when he said his party had lost more seats than it had won.
"We won in some regions, states and divisions, but also lost in some others," Htay Oo, the Union Solidarity and Development Party's acting chairman, told local TV station DVB. "We have (a) higher percentage of losses than wins." He himself had failed to win his seat, he said.
In early results from Sunday's vote -- billed as the country's freest in a generation -- the National League for Democracy had won 78 of 88 seats declared so far in the lower house of parliament. Hundreds more results, including many from remote areas with poor infrastructure, need to be announced before the full picture becomes clear.
The ruling party of President Thein Sein -- a former general who has overseen a series of political reforms in recent years -- had won only five of the seats declared so far, election officials said.
NLD supporters gathered amid jubilant scenes outside the party's Yangon headquarters Monday night, in anticipation of a historic, landslide victory in Myanmar
, which is also known as Burma.
Music played as they waved flags bearing the NLD's golden peacock emblem; many wore T-shirts emblazoned with pictures of Suu Kyi.
"We believe we can win," Ayea Nyeian Thu, a doctor, told CNN at the rally. "We don't want to see a military government any longer."
NLD supporters had been waiting a long time for a democratic vote, she said. "We want to celebrate."
The landmark election is seen as a test of the powerful Myanmar military's willingness to let the country continue along a path toward full democracy, after decades of military-dominated rule.
Thein Sein has promised that the outcome of Sunday's vote will be respected, but the system is already configured strongly in favor of the military, which gets to appoint a quarter of all lawmakers in the two houses of parliament.
That means the NLD would need to win more than two-thirds of the remaining seats in each house to secure majorities.
The public is electing 168 of the 224 representatives in the upper house of the national parliament, with the remaining quarter of seats reserved for lawmakers appointed by the military.
In the lower house, 325 of the 440 seats are up for grabs. Another 110 are reserved for military appointees, while voting has reportedly been canceled in the remaining five electable lower house seats because of security concerns.
Free and fair?
The changes ushered in under Thein Sein since 2011 have helped reduce the country's international isolation, with Western sanctions being eased and foreign investment starting to ramp up.
But human rights groups have warned more recently of a rise in politically motivated arrests as well as discrimination directed against the Muslim minority, notably the stateless Rohingya population
Questions have come up over how free and fair the current election will turn out to be. Suu Kyi, a Nobel Peace Prize laureate, expressed concern last week about irregularities in advance voting, fraud and intimidation.
Many people still remember the last national election her party contested, in 1990, which it was widely considered to have won. But the military rulers annulled the results and placed Suu Kyi and many of her colleagues under arrest.
Suu Kyi, the daughter of an independence leader, spent much of the next two decades under house arrest, becoming an internationally recognized symbol of democracy and the country's most popular politician.
'More openness and transparency'
Some observers have questioned the impartiality of the Union Election Commission, the body that oversees the vote, which has ties to the ruling USDP.
Daw Thein Thein Tun, an official from the commission, insisted Sunday that this election was much better than parliamentary elections in 2010, which were boycotted by the NLD.
"There are more people this time compared to 2010," she told CNN.
"There is more regulation, and this time there is more openness and transparency," she added. "You see the voting is free and fair."
But NLD candidate Nay Phone Latt was skeptical. He told CNN that the party had monitored some irregularities and had noted minor incidents of violence and attempted voter fraud.
He added that the likelihood of foul play would be strongest in remote rural areas.
Hundreds of thousands of people in Myanmar are disenfranchised, including Rohingya Muslims in the west of the country, who are denied citizenship, and residents of conflict zones where the election commission canceled voting.
Suu Kyi barred from presidency
After the outcome of the parliamentary vote is decided, lawmakers will begin the complex process of choosing a president.
Suu Kyi, who was elected to parliament in a by-election in 2012 and is seeking re-election for her seat this year, is barred from the presidency by the military-drafted constitution, which prohibits anyone with foreign family members from assuming the top office. Suu Kyi's late husband was British, and her two sons have British passports.
Suu Kyi said last week she would be "above the President" if her party won the parliamentary election.
Complicating any efforts to change the rules in the future, the military also has an effective veto over any proposed constitutional changes.
In spite of the political maneuvering that may lie ahead, U.S. Secretary of State John Kerry
congratulated the country on holding the election Sunday.
"Millions of people from around the country, many of whom were voting for the first time, seized this opportunity to move one step closer to a democracy that respects the rights of all -- a testament to the courage and sacrifice shown by the people of Myanmar over many decades," he said in a statement.
But Kerry added that the election was "far from perfect," noting "important structural and systemic impediments to the realization of full democratic and civilian government."