Brantly, pinch-hitting for Heath Bell (3-5), drew a one-out walk from Cory Gearrin (0-1) in the 10th and went to second when Gearrin hit Gorkys Hernandez with a pitch.
After Donovan Solano struck out, Reyes (3-for-5) blooped a single to shallow right-center field for the game-winning RBI.
Bell earned the victory with a 1-2-3 top of the 10th and John Buck launched a two-run shot for Miami, which evened its three-game series.
The Braves forced extra innings when they rallied from a 3-0 deficit in the ninth, highlighted by Brian McCann's two-run double.
Dan Uggla also drove in a ninth-inning run for the National League wild card-leading Braves.