"This is not our triumph, this is a triumph of the people," Suu Kyi said as she arrived at the headquarters of the National League for Democracy in Yangon to meet with fellow candidates and other party members.
The prediction of victories in 43 of the 44 seats where the NLD fielded candidates was based on the party's own estimates, according to party member Thae Da Win Aung.
It was still unclear whether the NLD had won the 44th seat, she said.
The official election results of Sunday's elections are expected sometime this week.
The NLD had already claimed victory on Sunday for Suu Kyi in the constituency of Kawhmu, south of the former capital city of Yangon, where she ran against a former military doctor.
While control of the parliament will not change even if the opposition officially wins all 44 seats, the vote itself marks an important step forward for many in the country who have lived under military rule for 50 years.
Suu Kyi, 66, led her party to a landslide victory the last time Myanmar held multiparty elections, in 1990. But the junta ignored the results and placed her under house arrest.
Released in November 2010, Suu Kyi was allowed to crisscross the country to rally support for the NLD for Sunday's race.
The NLD fielded a candidate for every one of the 45 seats up for grabs. But the election commission rejected one candidate apparently because his parents had foreign residency. The NLD has said it plans to challenge his exclusion.
The government had promised the vote would be free and fair, and allowed international observers to monitor the polling.
On Sunday, the NLD said it had received more than 50 reports of voting irregularities.
In one area, ballot sheets had wax placed over the check box for the NLD, making it easier to erase the mark later and annul the vote, the NLD said. In another area, ballots were found that had already been filled out, it said.
Suu Kyi said her party planned to file a complaint about the alleged infringements -- not out of a spirit of vengeance but to ensure the system works better next time.
Nay Zin Latt, an advisor to Myanmar President Thein Sein, said the voting took place without incident.
"If it is not free and fair, NLD would not win a majority," he said.
Suu Kyi has said she has no regrets about taking part in the by-elections because the process has raised people's political awareness.
In front of hundreds of supporters and journalists gathered at the NLD headquarters Monday, Suu Kyi said she planned to push for more emphasis on the role of the people in governing the country. She said she would happily work with anybody who wanted national reconciliation.
Myanmar's legislature has 664 seats, more than 80% of which are still held by lawmakers aligned with the military-backed ruling party, Union Solidarity and Development Party (USDP).
The 45 seats under contention in Sunday's elections were vacancies created by the promotion of parliamentarians to the Cabinet and other posts last year.
Still, the election was an opportunity for voters to weigh in during a time of enormous change in Myanmar, a country also known as Burma.
Analysts said it was the first real test of the government's commitment to transition from military rule.
Two years ago, Myanmar staged a general election that was widely derided as a sham.
Several former military leaders formed the Union Solidarity and Development Party (USDP) at the time to contest the election. Suu Kyi's party boycotted it.
After attracting international condemnation for manipulating the voting process in the 2010 race, Myanmar's leaders appear to have concluded that a fairer election will be proof to the world that the authorities can conduct a legitimate vote, experts said.
In the past 12 months, the country pardoned hundreds of political prisoners, secured a cease-fire with Karen rebels and agreed to negotiate with other ethnic rebel groups. Freer press rules have encouraged the proliferation of journals and magazines.
Myanmar's efforts to thaw its frosty relations with the rest of the world have been warmly welcomed and rewarded. In recent months, a steady procession of foreign ministers has visited the country and, in February, the European Union lifted a travel ban on Myanmar officials.
There have been hints, too, that a successful vote on Sunday would lead to the relatively swift unraveling of sanctions that have long choked the country's economy.
Thousands of Burmese living in exile around the world were watching the election for a clear sign that it is safe to return home.
As a member of parliament, Suu Kyi would be expected to be free to travel outside Myanmar -- and more importantly to return -- something that wasn't possible during her long years of repression and confinement.