While the results haven't been officially certified, the win by the National League for Democracy means it will choose the country's next President.
The announcement, made by the election commission Friday, comes on the fifth anniversary of Suu Kyi's release from house arrest
, where the 70-year-old had been for the better part of 20 years.
Sunday's elections were the first freely held in the nation in 25 years. Suu Kyi herself was reelected to her seat in the Kawhmu constituency in Yangon.
Despite her party's win, the leader of Myanmar's long-fought democracy movement can't become President.
A change in Myanmar's constitution, drafted by the military, prevents anyone with foreign family members from becoming the nation's leader.
Suu Kyi's late husband was British. Her children hold British passports.
Still, she's pushing forward.
She said last week she would be "above the President" if her party won the parliamentary election.
But complicating any efforts to change the rules in the future, the military also has an effective veto over any proposed constitutional changes.
Military still holds many of the cards
It will take at least another week to tabulate all the results, a presidential spokesman said.
The landmark election is seen as a test of the powerful military's willingness to let the country continue along a path toward full democracy, after decades of military-dominated rule in Myanmar, also known as Burma.
The ruling party of Thein Sein has promised the outcome of the vote will be respected, but the system is configured strongly in favor of the military, which gets to appoint a quarter of all lawmakers in the two houses of parliament.
The changes ushered in under Thein Sein since 2011 have helped reduce the country's international isolation, with Western sanctions being eased and foreign investment starting to ramp up.
But human rights groups have warned of a rise in politically motivated arrests as well as discrimination directed against the Muslim minority, notably the stateless Rohingya population
Suu Kyi, the daughter of an independence leader, expressed concern last week about irregularities in advance voting, fraud and intimidation.
She was widely considered to have won the last national election her party contested, in 1990, but the military rulers annulled the results and placed Suu Kyi and many of her colleagues under arrest.
She spent much of the next two decades under house arrest, becoming an internationally recognized symbol of democracy and the country's most popular politician.
After the outcome of the parliamentary vote is decided, lawmakers will begin the complex process of choosing a President.
U.S. Secretary of State John Kerry
has described Sunday's elections as "an important step forward" but also cautioned that what happens next is key to Myanmar's future.
"A peaceful post-election period is crucial for stability and maintaining the confidence of the people in the credibility of the electoral process and the overall political transition," he said in a statement.