With more than 98% of the votes counted, Weah led the Unity Party's Boakai 61.5% to 38.5%, according to the National Elections Commission (NEC)
Weah, the candidate for the Coalition of Democratic Change, won the first round of the election in October with 38.4% while Boakai netted 28.8%, according to the commission.
NEC Chairman Jerome Korkoya said in a press conference Thursday that the final election results would "hopefully" be announced on Friday.
Some Liberians took to the streets, chanting victoriously and dancing.
"We started the struggle and we are about to end the struggle in 2017," Nablah Washington, a Weah supporter, told Reuters.
"That is on December 26, we went to the ballot box and today we are seeing results and results that are coming out of all the 15 counties, they are communicating to us and showing us the road of victory," Washington said. "That is why we are rejoicing."
One of Weah's former clubs, Paris Saint-Germain, tweeted its congratulations.
"We knew George Weah way before he became President-elect of Liberia," the tweet reads. "Congrats to the PSG and world football legend on the latest chapter of his brilliant career!!!"
The runoff election was originally scheduled for November 7, but that vote was delayed when claims of fraud and "gross irregularities" were filed by Boakai's party. The complaints were dismissed.