The Marlins trailed 6-4 in the sixth inning but then reeled off the game's next seven runs to grab control.
Ozuna provided a two-run triple in the first inning and his two-run double in the sixth pulled Miami into a 6-6 tie.
Dobbs' clout in the eighth wrapped up the scoring for the Marlins, who banged out 16 hits in completing a three-game series sweep of New York.
Chris Coghlan and Ed Lucas each contributed four hits and a pair of RBI in the victory.
Wade LeBlanc (1-5) earned the decision with 1 2/3 scoreless frames in relief of starter Kevin Slowey, who had coughed up six runs on six hits over 4 1/3 innings.
Mets reliever Scott Rice (3-4) was tagged with the loss after surrendering four walks and three runs in a 2/3-inning stint.