It was not immediately clear if the incident, which took place outside Boakai's party headquarters, was deliberate or an accident.
A video of the aftermath shared on Facebook showed badly injured people sprawled over the road and pavement outside the Unity Party office.
Boakai, 78, a former vice president, narrowly defeated President George Weah in the November 14 presidential run-off, securing 50.64% of the votes cast, according to the election commission's final results.
Weah, a global soccer legend, conceded defeat to Boakai on Friday, ending a presidency where many have grown disillusioned with the lack of improvement in fields such as food security, employment and electricity supply.
The election has passed largely without incident and will mark only the second democratic transfer of power in Liberia in more than seven decades - the first was when Weah swept to power six years ago.
A Unity Party security guard said he waved down the car to stop. The driver then turned off his headlights and plowed into the crowd.
Two staff at the John F. Kennedy Medical Center said three people were in a critical condition.
"We did not hear anything because the music was playing..."
"I don't really know where the car came from. I just found myself on the ground," said Bendu Kiadii in the JFK trauma ward.
A police spokesperson did not immediately respond to a request for comment.
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