The National League for Democracy (NLD) needs 322 seats in total to form a government
Myanmar authorities are due to release early election results on Monday after what appeared to be an enthusiastic turnout despite fears of the novel coronavirus for a vote that is expected to give leader Aung San Suu Kyi a new term.
Sunday's general election was seen as a referendum on the fledgling democratic government led by Suu Kyi's ruling National League for Democracy (NLD), which remains popular at home but has seen its reputation collapse overseas amid allegations of genocide against the Rohingya Muslim minority.
The Election Commission is expected to announce early official results later on Monday.
In a running tally on its official Facebook page, the NLD said it had won 15 of the 315 seats being contested in the 425-member lower house.
"We believe that we will win, to the extent we can form a government," NLD spokesman Myo Nyun told Reuters by telephone.
The election had been peaceful and smooth, he said.
Representatives are also being elected to the upper house with 161 seats up for grabs in the 217-seat chamber.
The military, which ruled Myanmar for nearly 50 years until it began withdrawing from civilian politics in 2011, controls a quarter of seats in both houses of parliament, under a constitution it drew up and which Suu Kyi and her allies want to amend.
The NLD needs 322 seats in total to form a government and is expected to win again but with a smaller margin as new parties emerge and ethnic minority parties gain support in some regions.
In contrast to the wave of optimism that greeted the NLD's landslide win in 2015, Myanmar went into this election facing a surging Covid-19 outbreak, economic hardship and escalating ethnic conflicts.
Although Myanmar is seeing an average of 1,100 new coronavirus cases a day - compared with a handful in early August - fears of the virus appeared not to dampen Sunday's turnout among the 37 million registered voters.
The Election Commission has yet to release data on turnout but in the biggest city, Yangon, long lines of voters wearing face masks and shields formed from dawn.
But more than a million people across the country were unable to vote after polls were cancelled due to insurgencies.
Hundreds of thousands of Rohingyas, confined to camps and villages in Myanmar's Rakhine State, most without citizenship, were also unable to vote.
The Democracy and Human Rights Party, a Rohingya party, said in a statement it was "utterly disappointed" that its people had been disenfranchised.
The election commission has said the polls in areas affected by conflict had to be cancelled for safety reasons and that only citizens were entitled to vote.
Most Rohingyas are not considered Myanmar citizens but migrants from neighbouring Bangladesh even though many can trace family roots back many generations.
The United Nations has said there was genocidal intent in a 2017 army crackdown that forced 730,000 Rohingyas to flee to Bangladesh.
Myanmar rejects that saying its security forces were carrying out legitimate operations against Rohingya militants.