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.
Analysis
With the Kawhmu result close to a foregone conclusion, thoughts are already turning to what sort of a local parliamentarian Ms Suu Kyi will be.
"We need better transportation and opportunity for young people here," said U Myo Khine, a father of two, as he watched her convoy pass by.
Others have their eyes on the much greater prize, the general election of 2015.
"The army has changed and are now more lenient," said NLD official Myo Win. "So there is more of a possibility that Aung San Suu Kyi can become president in 2015."
BBC correspondent Rachel Harvey says the NLD alleged some voting irregularities in the capital, Naypyidaw.
A NLD spokesman told AFP news agency he had sent a letter of complaint to the election commission over allegations ballot forms had been tampered with.
Nyan Win said there had been complaints that wax had been put over the check box for the party, which could later be rubbed off to cancel the vote.
"This is happening around the country. The election commission is responsible for what is occurring," he said.
Burma's current government is still dominated by military and ex-military figures from the old regime that ruled the country for decades and was accused of widespread rights abuses.
But since 2010, when a transition to a new generation of leaders began, the government has impressed observers with the pace of change.
Most political prisoners have been freed, media restrictions have been relaxed and, crucially, Ms Suu Kyi and the NLD have been persuaded to rejoin the political process.
They have taken no part in Burma's political process since 1990, when the NLD won a landslide victory in a general election but the military refused to accept the result.
Ms Suu Kyi spent much of the following 20 years under house arrest and refused to take part in the 2010 election, which ushered in the current reforms.
The NLD is one of 17 opposition parties taking part in Sunday's election. Only a fraction of seats are up for grabs and the military-backed party will still dominate.
'Not irreversible'Ms Suu Kyi, 66, is standing for a lower house seat in the Kawhmu Township constituency outside Rangoon.
On Sunday, Ms Suu Kyi visited polling stations in Kawhmu before heading back to Rangoon.
The BBC's Fergal Keane, who is travelling with her, tweeted that she is still feeling the physical strains of the campaign. Last week Ms Suu Kyi suspended her campaign because of ill-health which aides said was triggered by exhaustion.
Burma's by-elections
- At least 45 seats are being contested by 176 candidates from 17 parties, with eight independents
- The Lower House has 440 seats (330 elected), the Upper House 224 seats (168 elected) and the regional assemblies 14, with 25% of the seats appointed by the military
- Aung San Suu Kyi is seeking a seat in Kawhmu district south of Rangoon
- Her party, the National League for Democracy (NLD), is contesting all seats
- By-election fills vacancies of those elected in 2010 polls who became ministers and deputy ministers
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.
"Still we are determined to go forward because this is what our people want," she said.
A small number of representatives from the Association of South-East Asian Nations (Asean), along with the EU and US, have been invited to observe polling.
More than 100 foreign journalists are believed to have received permission to cover the vote.
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.
"Political freedoms and economic freedoms always go together."
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