Suu Kyi's National League for Democracy party claimed to have won all 44 seats it contested in the 664-seat Parliament dominated by the military.
The country's election commission must confirm the results within a week.
The more than 150 observers from the United States, European Union, Canada, Australia, India and the Association of Southeast Asian Nations monitoring the polling reported some irregularities, but said the vote was an improvement over the November 2010 general election in which candidates allied with the military junta swept to victory.
The United States, in a statement issued by the White House press office, congratulated the people of Myanmar and Suu Kyi for their roles in Sunday's election process.
"This election is an important step in [Myanmar's] democratic transformation," the statement said, "and we hope it is an indication that the government of [Myanmar] intends to continue along the path of greater openness, transparency, and reform."
If confirmed, the NLD victory would include four seats representing the capital, Naypyitaw, overwhelmingly populated by civil servants and until now considered a ruling military party stronghold.
The opposition victory in Myanmar, formerly known as Burma, would suggest disaffection within the ranks of government, The New York Times reported.
Suu Kyi, a 66-year-old Nobel Peace laureate who spent 15 years under house arrest, won a seat in the 440-seat lower house, defeating a military doctor from the junta-backed Union Solidarity and Development Party.
Clinton -- who met with Suu Kyi and the country's new president, U Thein Sein, Dec. 1 -- gave cautious support to Sunday's elections.
"The United States congratulates the people who participated, many for the first time, in the campaign and election process," Clinton said in Istanbul, Turkey, at an international conference on providing aid to the Syrian opposition.
"It is too early to know what the progress of recent months means and whether it will be sustained," Clinton said. "There are no guarantees about what lies ahead for the people of Burma. But after a day responding to a brutal dictator in Syria, who would rather destroy his own country than let it move toward freedom, it is heartening to be reminded that even the most repressive regime can reform and even the most closed society can open."
In January, Clinton said Washington would send an ambassador to Myanmar, located on the Indonesian peninsula in Southeast Asia, for the first time in more than two decades.
The Obama administration identified Sunday's poll as a critical measure of Myanmar's progress, suggesting it may begin to dismantle a raft of economic sanctions against the regime -- moves that are widely expected to happen in stages, The Washington Post reported.
The European Union also said the fairness of Sunday's vote would crucial in determining whether it would lift economic sanctions against the country.
A representative for U.N. Secretary-General Ban Ki-moon said Ban recognized "the courage and vision" of Thein Sein, which made possible such progress.
"The secretary-general takes note of the constructive efforts and positive statements by the government and other key stakeholders, including [Suu Kyi], in the lead-up to and following the by-election," Ban's representative said. "The secretary-general urges the government and all political forces to cooperate and build on the important achievement of the by-election and to consolidate the country's democratic gains. He also calls on all of [Myanmar's] stakeholders to redouble their efforts toward genuine national reconciliation and long-term and durable peace."
Under Thein Sein's presidency, the country of 60 million has undergone a transformation that has seen cease-fires declared with ethnic rebel armies, censorship laws eased, political prisoners freed and Suu Kyi released from house arrest with permission for her party to participate in the election.