Preliminary figures showed the incumbent with 62%
compared to 34% for his closest competitor, opposition leader Kizza Besigye. But Besigye's party, the Forum for Democratic Change, rejected the results and demanded an independent audit of the elections.
Besigye is under house arrest, with no one is allowed in or out to see him.
Elections officials are scheduled to announce the final results sometime later this weekend.
The country had a voter turnout of 63%, according to the election commission.
More than 60% of the vote
As the nation awaited the official results, Besigye's arrest and the deaths of two people at his opposition campaign office added to the tension. Besigye was put under "preventative arrest" Friday at his home in Kampala, along with six officials from his party, police said.
Authorities detained the officials because they planned to announce unauthorized results tallied by the opposition, police spokesman Patrick Onyango said.
Under law, only election officials may announce results, he said.
The offices of the opposition party, where two people died following violence, had been besieged by police and military forces, the Red Cross said.
Incumbent in power since 1986
Museveni, 71, has been in office since his rebel group seized power in 1986. In 2005, the constitution was changed to allow him to extend his time in office.
As he tries to maintain his grip on power, experts emphasized the difficulty of unseating incumbents in Africa.
"A re-election for Museveni would signal the persistent advantages incumbents have in controlling the political process, making it very difficult for opposition parties or candidates to compete with national structures, finance and support from partisan government institutions," said Magnus Taylor, an analyst at the International Crisis Group.
Besigye, 59, is one of eight candidates running for President this year.
He is also Museveni's former doctor and served as a minister in his Cabinet.
This is not Besigye's first attempt to unseat his former boss. He lost presidential bids in 2001, 2006 and 2011.
U.S.: 'The Ugandan people deserve better'
Kenyan President Uhuru Kenyatta was among those who reacted to Saturday's announcement, tweeting that he was "very pleased to congratulate ... Museveni."
"The people of Uganda have spoken, and they have spoken very clearly," said Kenyatta, who has been in office since 2013. "We respect their choice."
The U.S. government commended Ugandans for peacefully turning out to vote, though a statement from State Department spokesman Mark Toner did not mention Museveni by name, much less congratulate him.
Toner's statement did mention Besigye's arrest and called for his immediate release. The U.S. spokesman also pointed to a number of other issues, such as "delays in the delivery of voting materials, reports of pre-checked ballots and vote buying, ongoing blockage of social media sites and excessive use of force by the police."
All these things, if true, "are deeply inconsistent with international standards and expectations for any democratic process," Toner said.
"(These issues) collectively undermine the integrity of the electoral process," the spokesman added. " The Ugandan people deserve better."