President Akufo-Addo has congratulated Liberia's president-elect, George Weah.
He beat 73-year-old Joseph Boakai in the presidency race.
The victor will replace Ellen Johnson Sirleaf, Africa's first female president, who is stepping down after two terms in office.
He will rather compound the problems in Liberia because I don't see him being trusted by the President-Elect, whom he (Obasanjo) openly worked against in 2005 to impose the outgoing President, Ellen Johnson Sirleaf for reasons best known to him.
The runoff was triggered since neither Weah nor Boakai managed to win 50 percent of the vote, which would have handed either candidate an automatic victory.
Weah supporters paraded through the streets of the capital Monrovia and honked vehicle horns to celebrate the news.
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Vote counting has begun in Liberia following a peaceful run-off election for a new president on Tuesday, pitting former global footballer George Weah against Vice-President Joseph Boakai. His failure to secure an outright majority forced a run-off.
Weah, a former football star and senator from the Congress for Democratic Change party, has won more than 61 percent of the vote with 98 percent of ballots counted, according to Liberia's election commission.
Turnout was "nowhere near that of the first round held in October", our correspondent said, attributing the low numbers to a seven-week delay in holding the runoff.
Sirleaf is stepping down after two terms as president.
Weah has also fended off barbs over his vice-presidential pick, Jewel Howard-Taylor, the ex-wife of jailed former president and warlord Charles Taylor.
Taylor is serving his 50-year prison sentence in the United Kingdom for war crimes related to the conflict in neighbouring Sierra Leone.