Her National League for Democracy (NLD) competed for all 45 seats, in the first vote it has contested since 1990.
Ms Suu Kyi’s supporters said unofficial results showed her set to win her seat.
The elections are being seen as a key test of Burma’s pledge to political reform, though the military-backed governing party will remain dominant.
Foreign journalists and international observers are being given the widest access they have ever had in Burma.
The European Union hinted that it could ease some sanctions if the vote went smoothly.
“We hope the whole day can be run in a peaceful way and we’ll make an evaluation later on the basis of all the polling sessions that we will be seeing,” EU observer Ivo Belet said.
BBC correspondent Rachel Harvey says the NLD alleged some voting irregularities in the capital, Naypyidaw.
Earlier, Ms Suu Kyi described this year’s election campaign as not ”genuinely free and fair” and warned that reforms were “not irreversible”.
But she said she and the NLD did not regret taking part.
EU Trade Commissioner Karel De Gucht said political sanctions on Burma were mostly “aimed towards individuals” and could be eased when EU foreign ministers met in Brussels on 23 April.
The lifting of such sanctions could “even happen with immediate effect”, he told AFP news agency.
“I am excited by the prospect that finally, hopefully, Myanmar [Burma] citizens will get more freedom,” Mr De Gucht added.