Paulino, in a pinch-hitting role with none out and the bases loaded, drove a pitch from George Sherrill (0-1) off the center field wall, plating two runs for the Marlins and forging a 6-6 tie.
The Dodgers then intentionally walked Hanley Ramirez to load the bases again and Cantu followed by hitting a fly ball to center, enabling Chris Coghlan to tag up and score the winning run.
Florida also benefited from Gaby Sanchez's three-run homer. Starter Josh Johnson got a no-decision, allowing three runs on eight hits with seven strikeouts in five innings while Jose Veras (1-0) picked up the win.
The Dodgers' Matt Kemp went 2-for-5 with a solo homer, two runs batted in and two runs scored in the loss.