Boakai led with 50.89% of the vote over Weah's 49.11%, according to results from 99.6% of polling stations, the country's elections commission said on Friday.
Final results are expected in the coming days.
Boakai, a softly spoken career politician, was more than 28,000 votes ahead of Weah, according to the latest results.
Arkoi Sarkor, 43, told Reuters she supported Boakai because she was not able to get a job during Weah's term.
"We can safely declare victory – Boakai is the next president, but we will wait for the final result before celebrating," Boakai's spokesperson Amara Konneh said in an interview.
Persons:
Unity Party Joseph Boakai, Carielle, Joseph Boakai, George Weah, Boakai, Arkoi Sarkor, Boakai's, Amara Konneh, Carielle Doe, Alphonso Toweh, Anait Miridzhanian, Edward McAllister, Louise Heavens, Jonathan Oatis
Organizations:
Unity Party, Reuters, Boakai, Thomson
Locations:
Monrovia, Liberia, MONROVIA, West, Weah