Johan Camargo led off the 10t with a single, advanced on a sacrifice and scored when Phillips lined a ball up the middle off A.J. Ramos (1-3).
Nick Markakis' RBI double in the ninth sent the game into extra innings. Arodys Vizcaino (3-2), the seventh Atlanta pitcher, faced four batters in the 10th to get the win.
The Braves had lost five of seven overall and eight of 11 at home, but they overcame an uncustomary rocky start by pitcher Jaime Garcia to cool off the Marlins.