Miami overcame a 3-1 deficit, tying the contest in the sixth when Logan Morrison tripled in a run and then scored on an infield single by Ozuna.
The Marlins then batted around in the eighth and scored three times after two were out. Ozuna singled home the tiebreaking run, Greg Dobbs also had an RBI single and Ozuna scored on a throwing error by shortstop Joaquin Arias.
The eighth-inning surge gave the win to A.J. Ramos (1-2), the third of five Miami pitchers who worked the seventh inning. It was the first victory of his major league career, which consists of 40 relief appearances over two seasons.
Steve Cishek put two runners on with one out in the ninth, but escaped further problems to grab his 12th save.
Jake Dunning (0-1), competing in his fourth major league game, took the loss after giving up the first of Miami's eighth-inning runs.