In an address to the nation he said “the Liberian people have spoken and we have heard their voice”.
The opposition candidate holds an unassailable lead of 28,000 votes with nearly all ballots counted.
A former football star, President Weah has been in power since 2018. He will step down in January.
He came into the job on a wave of enthusiasm, especially from younger voters, having won that election – also against Mr Boakai – by a large margin.
But a perception that he had failed to tackle corruption, rising prices and continued economic difficulties tarnished his image.
Mr Weah was magnanimous in defeat, beginning his five-minute address by saying he had “the utmost respect for the democratic process that has defined our nation”, adding that he had spoken to Mr Boakai who he called the “president-elect”.
Earlier the electoral commission announced that Mr Boakai, a 78-year-old political veteran had 50.89% of the votes, while President Weah had 49.11%.
The president referred to the closeness of the race saying it “reveals a deep division within our country” and called on Liberians to “work together to find common ground… unity is paramount for mama Liberia”.
A long period of civil war in which an estimated 250,000 people died ended just 20 years ago.
When the latest set of results were announced earlier on Friday, they sparked celebrations in the capital, Monrovia.
Mr Boakai’s supporters gathered at his party’s headquarters in the city, convinced that he had won the election.
They called for President Weah to go, and chanted “we beat the Buga dancer” – referring to a song that became associated with his campaign.
The electoral commission says it has announced the results from 99.58% of the polling stations following Tuesday’s run-off election.
The run-off between Mr Boakai and Mr Weah was triggered after neither candidate got more than 50% of the vote in last month’s first round. There were 18 other candidates.
In that vote, the president got the largest share and was just 7,000 votes ahead of Mr Boakai.
The poll was the tightest presidential contest in Liberia since the civil war.
Mr Boakai’s campaigning focused on the need to rescue the nation from what he called “mismanagement” by Mr Weah’s administration.
The president dismissed Mr Boakai’s allegations, saying he had made significant strides, including introducing free tuition for university students.
This is the fourth time a presidential election has taken place since the war ended.
Observers from the regional bloc, Ecowas, deemed the run-off largely peaceful, AFP news agency reports.
But there were isolated incidents that led to “injuries and hospitalisations” in the provinces of Lofa, Nimba, Bong and Montserrado, it quotes observers as saying.
