John Buck drew a walk from Anthony Varvaro (0-2) to lead off the 12th. Pinch-runner Brett Hayes went to second on a Bryan Petersen sacrifice and the Braves then issued an intentional walk to Donnie Murphy.
One out later, Cristhian Martinez came in to pitch and Stanton greeted him by grounding a 1-2 pitch past diving shortstop Jack Wilson, plating Hayes with the go-ahead run.
Clay Hensley (4-6) earned the win with two shutout relief innings and Leo Nunez racked up his 35th save in the bottom of the 12th as the Marlins took the opener of a three-game series.
Alex Gonzalez went 3-for-4 and drove in the game-tying run with an eighth-inning single for the National League wild card-leading Braves, who lost for the seventh time in eight games.
They remained 4 1/2 games ahead of St. Louis, however, after the Cardinals lost to Pittsburgh Monday.
Swim Week Miami Beach 2014 [PHOTOS]