Myanmar's opposition National League for Democracy says it is confident of victory in the first openly contested national election in 25 years.
An NLD spokesman said it expected to win about 70 percent of seats. Party leader Aung San Suu Kyi said: "I think you all have the idea of the results."
Official results have been released for just 12 seats, all won by the NLD.
The military-backed Union Solidarity Development Party (USDP) has been in power since 2011.
"We are on track to win more than 70 percent of seats around the country, but the election commission has not officially confirmed yet," NLD spokesman Win Htein told AFP news agency. The 12 seats announced so far are all in Yangon.
The acting chairman of the USDP has told BBC Burmese that he has lost his own seat in the constituency of Hinthada to the NLD - seen as a key indicator of election results.
"We have to find out the reason why we lost," U Htay Oo said. "However, we do accept the results without any reservations. We still don't know the final results for sure."
Earlier, Ms Suu Kyi addressed a crowd at the NLD's headquarters in Yangon, urging them to be patient.If the NLD wins two-thirds of the contested parliamentary seats, it would end decades of military control.
A quarter of the parliamentary seats are reserved for the army, and for the NLD to have the winning majority it will need at least two-thirds of the contested seats.
The BBC's Jonah Fisher in Yangon says that while this is a very big ask, it is by no means impossible if the party, which is popular in urban areas, manages to win seats in rural areas which tend to be dominated by ethnic minorities.
But Ms Suu Kyi cannot become president because the constitution bars anyone with foreign children from holding the post. Her two sons, with her late husband, are British.
If the NLD win, it will face difficulties in changing the constitution on its own as the document still gives the military considerable power, and the party would most likely nominate someone else to be president. Ms Suu Kyi has said she would be "above the president".
Tens of thousands of officials and volunteers have been counting the votes, first in each of the 50,000 polling stations, then tallying them in constituency offices of the Election Commission.
In one of the earliest and most significant known results, the ruling party's Shwe Mann, who is speaker of the lower house of parliament, conceded defeat in his constituency to the NLD candidate.
The full results will not be known for at least a few days, and the president will only be chosen in February - or possibly later.
International observers said the voting process was generally smooth, with some isolated irregularities.
But hundreds of thousands of people - including minority Rohingya Muslims - were denied voting rights, raising concerns about the fairness of the poll.
US Secretary of State John Kerry hailed the elections as a step towards democracy, but added that they were far from perfect.
About 30 million people were eligible to vote in Sunday's election in Myanmar, which is also known as Burma.
Turnout has been estimated at about 80 percent, in what were the first national elections since a nominally civilian government took power in 2011.
More than 6000 candidates from more than 90 parties were vying for parliamentary seats.
A quarter of the seats were reserved for unelected military representatives, who were expected to side with the USDP.
The NLD, led by former political prisoner and Nobel Peace Prize laureate Aung San Suu Kyi, therefore needs to win 67 percent of contested seats in order to gain a majority and be able to appoint the president.
Ms Suu Kyi cannot become president herself, because of a clause in the constitution which bars anyone with foreign children from holding the post. Her two sons, with her late husband, are British.
Despite this, she has insisted she will lead the country and be "above the president" if the NLD takes power.
Current President Thein Sein has said he will accept the election result.