President Weah said he had spoken to opposition leader Joseph Boakai to congratulate him on his victory.
Liberian incumbent president and football legend George Weah conceded defeat on Friday evening after nearly complete returns showed opposition leader Joseph Boakai leading with nearly 51 percent of the run-off ballots.
"The results announced tonight, though not final, indicate that... Boakai is in a lead that we cannot surpass," Weah said in a speech on national radio.
The "CDC (party) has lost the election, but Liberia has won. This is the time for graciousness in defeat, to put national interest above personal interest," he said.
The expected victory for former vice president Boakai would be sweet revenge for the 78-year-old, who lost to Weah, 57, by a large margin in the second-round presidential vote in 2017.
With more than 99.5 percent of the polling stations reporting vote tallies after Tuesday's second-round vote, Boakai had garnered 50.89 percent of ballots cast, with Weah on 49.11 percent, according to official results released by the electoral commission.
Boakai was 28,000 votes ahead of Weah, according to Friday's figures, after the two men finished neck-and-neck in the first-round last month, with a national lead of just 7,126 votes for Weah.
The United States extended its congratulations to "president-elect Boakai on his victory and President Weah for his peaceful acceptance of the results".
"We call on all citizens to follow President Weah's example and accept the results," US State Department spokesman Matthew Miller said in a statement.
Weah said he had spoken to Boakai "to congratulate him on his victory".
"The Liberian people have spoken, and we have heard their voice. However, the closeness of the results reveals a deep division within our country," Weah said in his speech.
"Let us heal the divisions caused by the campaign and come together as one nation and one united people."
The elections were the first since the United Nations in 2018 ended its peacekeeping mission, created after more than 250,000 people died in two civil wars in Liberia between 1989 and 2003.
International observers, including the European Union, commended Liberia this week for holding a peaceful election.
Regional bloc the Economic Community of West African States (Ecowas) said the election was "largely" peaceful, but it noted isolated incidents that led to "injuries and hospitalisations" in four provinces.
Clashes during the campaign left several dead before the first round and raised fears of post-election violence.
Around 2.4 million Liberians were eligible to vote on Tuesday, but no turnout figures have been released.
Boakai is an old political hand, having served as vice president to Ellen Johnson Sirleaf, Africa's first elected female head of state, from 2006 to 2018.
He is set to lead the English-speaking country of around five million people, one of the poorest in the world, for six years.
More than a fifth of the population lives on less than $2.15 a day, according to the World Bank.
Boakai forged alliances with local barons, including former warlord and senator Prince Johnson, who supported Weah six years ago.
Dozens of Boakai's supporters danced in celebration outside one of his party's offices in the capital Monrovia.