Before Fontenot's game-winner, Derek Lee hit a game-tying solo homer.
Alfonso Soriano had three hits, one a three-run homer in the fourth. Lee had two RBI and Aramis Ramirez added two hits and one RBI, helping the Cubs to a one-game lead over the Brewers in the NL Central Division before the start of a critical four-game series in Milwaukee starting Monday night.
The winner was Chad Gaudin (1-1).
Jeff Samardzija recorded his first career save.
Mike Jacobs gave Florida a 5-0 lead with a three-run homer in the second and a two-run shot in the third. He finished with three hits.
Dan Uggla had two hits, one a solo homer in the fifth, and Hanley Ramirez added two hits for the Marlins, who had won two in a row.
The losing pitcher was Mark Hendrickson (7-8).
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys