The National League for Democracy won 40 of the 44 seats that it contested, according to partial results announced by the National Electoral Commission on state television.
The party had claimed earlier Monday it won at least 43 seats -- including Suu Kyi's.
"This is not our triumph, this is a triumph of the people," Suu Kyi said as she arrived at the party's headquarters in Yangon to meet with fellow candidates and other party members.
While control of parliament will not change even if the opposition 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.
"The people were living in prison," said Myint Maung, a Yangon resident. "Aung San Suu Kyi held the key to open the door."
A White House statement Monday congratulated Suu Kyi and the people of Myanmar, also known as Burma.
"This election is an important step in Burma's democratic transformation, and we hope it is an indication that the government of Burma intends to continue along the path of greater openness, transparency, and reform," the White House statement said.
U.S. Secretary of State Hillary Clinton cautiously hailed the results of the voting Sunday, telling CNN that she was "very hopeful for the the people" of Myanmar.
U.N. Secretary-General Ban Ki-moon also commended the country "for the peaceful and largely orderly manner" in which the elections were held, according to his spokesperson.
The NLD prediction 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.
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 United States announced in January that it would exchange ambassadors with Myanmar after the regime released political prisoners.
Clinton visited Myanmar in December -- a historic trip marking the first time a secretary of state had been to the country in more than 50 years -- and British Foreign Secretary William Hague visited the following month.
On Sunday, Clinton said she had been impressed with her visit.
"I was very touched by the visit that I made and the commitments I received from members of the government who were quite sincere in their desire to move their country forward," she said, even as she underlined that the United States wanted to see "continuing progress."
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.
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, the 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.
Win Naing Kyaw wore a T-shirt Monday with the likeness of independence hero General Aung San, the father of Suu Kyi. "Like everyone, I am expecting democracy now," he said.
Daw Tin May Oo, 77, insisted Monday that the country was happy with the results. "The country has only just survived the past 20 years or so ... with its poverty, lack of food and broken lives," she said. "The situation has now turned around."
Analysts said the vote 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 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 media 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 February, the European Union lifted a travel ban on Myanmar officials.
There have been hints, too, that a successful vote 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.
