Pro-democracy leader Aung San Suu Kyi says she hopes Sunday's by-elections mark the start of a new era in Burma.
Calling the polls a "triumph of the people", she said the goal now was reconciliation with other parties.
According to partial official results, Ms Suu Kyi's party took at least 40 of the 45 seats being contested, state media say.
Although the vote is seen as a key test of political reform, the army and its allies still dominate the parliament.
The by-elections were being held to fill 45 parliamentary seats left vacant by the appointment of ministers after the polls that formally ended military rule in November 2010. There are 664 seats in parliament altogether.
Aung San Suu Kyi's National League for Democracy (NLD) was competing in its first elections since 1990, after boycotting the 2010 polls. It was one of 17 opposition parties that took part.
The European Union's Trade Commissioner, Karel De Gucht, told the BBC that that this result could lead to the EU easing sanctions on Burma.
"Her [Suu Kyi's] party has won the larger part of the seats, that I think is an indication that the elections have been free and fair and that the result reflects the will of the voters," he said.
'Triumph of people'
Ms Suu Kyi's comments came as she addressed a crowd of supporters outside NLD headquarters in Rangoon, Burma's commercial capital.
"It is not so much our triumph as a triumph of the people who have decided that they have to be involved in the political process in this country," she said. "We hope this is the beginning of a new era."
"We hope that all other parties that took part in the elections will be in a position to co-operate with us to create a genuinely democratic atmosphere in our nation."
Apart from winning her own seat, Ms Suu Kyi appears to have helped a number of her colleagues to victory, correspondents say.
Partial results appear to confirm NLD projections that the party has won almost all of the 44 seats it contested, including some in the remote capital, Nay Pyi Taw.
But even if the NLD wins most of the seats, the army and its proxy Union Solidarity and Development Party (USDP) will still hold about 80% of seats in parliament.
Ms Suu Kyi - who spent years under house arrest after her party won polls in 1990 but was not allowed to take power - has promised to use her voice to continue to push for further reform.
Speaking in Cambodia ahead of an Asean summit, Burma's Foreign Minister Wunna Maung Lwin said the polls had been "free, fair and transparent".
During the campaign, foreign journalists and international observers were given the widest access to the former military-ruled nation for years.
US Secretary of State Hillary Clinton congratulated Burma on holding the vote and said that the US was ''committed to supporting these reform efforts".
US lawmakers who drafted sanctions against Burma remained cautiously optimistic.
"While much remains to be done in Burma, Suu Kyi's apparent election to parliament, like that of the apparent election of large numbers of her NLD colleagues, is an important step forward for the country," said Senator Mitch McConnell.
Representative Joe Crowley said ''now is not the time'' to rush towards lifting the sanctions.
"Far too many political prisoners are still locked behind bars, violence continues against ethnic minorities and the military dominates not only the composition but the structure of the government," he added.