Calling the polls a “triumph of the people”, she said the goal now was reconciliation with other parties.
Official results for the polls that saw 45 seats contested are expected later this week.
But Ms Suu Kyi’s National League for Democracy (NLD) said she easily won in her Kawhmu constituency, and that the party expects to win multiple seats.
Aung San 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.”
Aung San Suu Kyi has promised to use her voice to push for further reforms.
NLD officials say they believe the party has won almost all of the 44 seats it contested, including some in the remote capital, Nay Pyi Taw. There has been no formal word yet from the Election Commission.
US Secretary of State Hillary Clinton congratulated Burma on holding the vote and said that the US was ”committed to supporting these reform efforts”.
The European Union hinted that it could ease some sanctions if the vote went smoothly.