The claim, made in an emailed statement by the Cambodia National Rescue Party, was impossible to immediately verify, as no results had yet been released by the authorities.
At a news conference after the statement was released, party leader Sam Rainsy was less clear about whether the claim was that his party had actually won a majority at the polls, stressing that it was a victory for the country.
There was no immediate reaction from the ruling Cambodian People's Party of Prime Minister Hun Sen, who has ruled Cambodia for nearly three decades.
Rainy has a reputation for provocative statements that have sometimes landed him in trouble. His claim, coming before the release of official results or any statements from the ruling party, could be seen as a pre-emptive move against Hun Sen's side stealing the election.
It also sets up the possibility of serious post-election confrontations. Hun Sen, in statements suggesting there would be chaos if he lost, had implied he would not accept an opposition victory.
"Today is a historical day when the CNRP won the parliamentary elections for the fifth mandate period in 2013," said the party's statement, emailed about hours after the polls closed at 3 p.M. "CNRP wishes to thank all compatriots heartily that voted for the CNRP so that the party won these elections."
"CNRP wishes to appeal to all compatriots to maintain calm and order to wait for the official election results that will emerge soon and appeal to all compatriots to avoid to use violence or to create any trouble that could cause social unrest," the statement said.
Sam Rainsy began his day with a visit to a polling station near his party's office in Phnom Penh. Hundreds of voters, particularly younger ones, greeted him enthusiastically, with some of the bolder ones hugging him.
Counting of the paper ballots at polling stations began shortly after the polls closed, with representatives of political parties witnessing the process.
