The balloting passed off peacefully on Tuesday, with more than 2.4 million voters called to choose between incumbent Weah, 57, who is seeking a second term, and 78-year-old Boakai.
Observers expect the result to be tight after the two candidates came roughly neck-and-neck in the first-round vote last month.
With results counted from 1,315 of the 5,890 polling stations across the country, Boakai was ahead with 50.7 percent of the votes, Davidetta Browne Lansanah, chairwoman of the electoral commission, told reporters.
She did not specify what proportion of the total number of voters these polling stations represented. They accounted for just over 386,000 votes.
In the first round vote, held last month, Weah led his old rival by just 7,126 votes nationally.
But in 2017's presidential race, he easily beat Boakai in the second round, taking more than 61 percent.
The electoral commission has 15 days from the polls to publish the results but could do so sooner.
The electoral commission is expected to report daily on the state of the count which began shortly after the polling stations closed on Tuesday evening.
No major incident'
The elections are the first since the UN in 2018 ended its peacekeeping mission, created after more than 250,000 people died in two civil wars in Liberia between 1989 and 2003.
Fears the polls could be marred by violence or voting irregularities and over the acceptance of the results loomed large.
Both national and foreign observers followed the balloting and only a few minor incidents have so far been reported.
"From our observation, the process went well," Oscar Bloh, head of the Election Coordinating Committee, a civil society group, told AFP.
"Overall the process was peaceful, there was no major incident that we observed, though we didn't cover all the polling centers."
He declined to comment on voter turnout.
The electoral commission chief reported "isolated incidents" during the collection of the votes and the counting process. including an assault on an election supervisor, who was injured.
The Economic Community of West African States, which sent observers, welcomed "the generally peaceful conduct of the elections so far".
ECOWAS in its statement however voiced "deep concern over provocative statements and alleged planned conferences by political actors to prematurely declare victory."
But the West African bloc did not specify to whom it was referring.
It also warned "individuals or groups that they would be held solely accountable for any acts that may lead to violence and undermine the hard-earned peace and stability of Liberia."
ECOWAS, of which Liberia is a member, can impose sanctions.
Clashes during election campaigning left several dead and had raised fears of post-election violence.
The campaigning was also marked by the propagation of disinformation.
Weah -- the only African to win football's most prestigious individual award, the Ballon d'Or -- is popular among young people but must defend his record in office, while Boakai is an old hand who has worked in both the public and private sectors.
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