In a statement, she urged supporters to show restraint in their celebrations.
The vote is a key test of political reforms, though the army and its allies dominate the 664-seat parliament.
The NLD was competing in its first elections since 1990.
Thousands of people who gathered outside the NLD headquarters in Rangoon danced and cheered at reports that Ms Aung San Suu Kyi had won her seat.
"It is the people's victory! We have taught them a lesson," one shopkeeper wearing an NLD T-shirt told the Associated Press news agency.
AnalysisWith the Kawhmu result close to a foregone conclusion, thoughts are already turning to what sort of a local parliamentarian Ms Aung San 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."
During the campaign, foreign journalists and international observers were given the widest access for years.
The European Union hinted that it could ease some sanctions if the vote went smoothly.
The BBC's Rachel Harvey in Rangoon says the NLD alleged some voting irregularities in the capital, Nay Pyi Taw.
NLD spokesman Nyan Win told AFP news agency he had sent a letter of complaint to the election commission over allegations ballot forms had been tampered with.
He said there had been complaints that wax had been put over the tick-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.
- A total of 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), contesting 44 seats
- By-election fills vacancies of those elected in 2010 polls who became ministers and deputy ministers
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 politics 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 was one of 17 opposition parties that took part in Sunday's election. Only a fraction of seats were up for grabs and the military-backed party will still dominate.
Ms Aung San Suu Kyi, 66, was standing for a lower house seat in the Kawhmu Township constituency, outside Rangoon.
On Sunday, she visited polling stations in Kawhmu before heading back to Rangoon.
Our correspondent says regardless of the outcome of the by-elections, Burma's balance of power will not change overnight and full democracy is still a long way off.
She says all Burma's ethnic minorities must be made to feel they have a voice in the political system and that President Thein Sein still needs to convince more wary parliamentarians that it is in the country's interests to open up to the outside world.