The National Election Commission said Wednesday the NDL has won 163 of the 182 seats declared so far in the Pyidaungsu Hluttaw, the country's parliament.
Confident in the returns, Suu Kyi told the BBC Tuesday
that the NDL is on a winning track.
"The results have been coming in steadily, and we probably will get between around 75% in the union legislature," she said, referring to the national parliament.
The military-aligned ruling party has admitted it has lost more seats than it has won. But the scale of its losses and the NLD's victories is still emerging.
The results on hundreds more seats have yet to come in.
NLD supporters gathered amid jubilant scenes outside the party's Yangon headquarters, in anticipation of a historic, landslide victory in Myanmar
, which previously was called Burma.
Music played as they waved flags bearing the NLD's golden peacock emblem; many wore T-shirts emblazoned with pictures of Suu Kyi.
"We believe we can win," Ayea Nyeian Thu, a doctor, told CNN at the rally. "We don't want to see a military government any longer."
The landmark election is seen as a test of the powerful Myanmar military's willingness to let the country continue along a path toward full democracy, after decades of military-dominated rule.
The ruling party of President Thein Sein has promised that the outcome of Sunday's vote will be respected, but the system is already configured strongly in favor of the military, which gets to appoint a quarter of all lawmakers in the two houses of parliament.
That means the NLD would need to win more than two-thirds of the remaining seats in each house to secure majorities.
The public is electing 168 of the 224 representatives in the upper house of the national parliament, with the remaining quarter of seats reserved for lawmakers appointed by the military.
In the lower house, 325 of the 440 seats are up for grabs. Another 110 are reserved for military appointees, while voting has reportedly been canceled in the remaining five electable lower house seats because of security concerns.
Free and fair?
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 more recently of a rise in politically motivated arrests as well as discrimination directed against the Muslim minority, notably the stateless Rohingya population
Questions have come up over how free and fair the current election will turn out to be. Suu Kyi, the daughter of an independence leader, expressed concern last week about irregularities in advance voting, fraud and intimidation.
Many people still remember the last national election her party contested in 1990; it was widely considered to have won that one. 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.
On Tuesday, she told the BBC that she doesn't expect a repeat of 1990 this time around.
"The times are different. The people are different," she said, describing citizens as "very much more alert to what is going on around them."
Nonetheless, hundreds of thousands of people in Myanmar are disenfranchised, including Rohingya Muslims in the west of the country, who are denied citizenship, and residents of conflict zones where the election commission canceled voting.
Suu Kyi barred from presidency
After the outcome of the parliamentary vote is decided, lawmakers will begin the complex process of choosing a president.
Suu Kyi, who was elected to parliament in a by-election in 2012 and is seeking re-election for her seat this year, is barred from the presidency by the military-drafted constitution, which prohibits anyone with foreign family members from assuming the top office. Suu Kyi's late husband was British, and her two sons have British passports.
Suu Kyi said last week she would be "above the President" if her party won the parliamentary election.
Complicating any efforts to change the rules in the future, the military also has an effective veto over any proposed constitutional changes.
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.