Aung San Suu Kyi`s National League for Democracy is on track for a majority, taking about 80% of declared seats.
But a quarter of parliamentary seats are allocated to the military, meaning it will remain hugely influential.
Last weekend`s vote was Myanmar`s first openly contested general election for 25 years.
The NLD won elections decisively in 1990 - only for the result to be nullified and Ms Suu Kyi placed under long-term house arrest.
President Thein Sein and Min Aung Hlaing have congratulated Ms Suu Kyi on her party`s performance in the polls, though have not formally conceded defeat. They have agreed to hold talks on national reconciliation with her after the final results are announced.
More than 47% of results have been declared so far, and on Thursday morning, the NLD were said to be 38 seats short of a majority.
The ruling military-backed Union Solidarity Development Party (USDP) - which won the last, widely criticised election five years ago - has so far gained about 5% of the seats being contested in Myanmar, also known as Burma.
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