Trailing 1-0 heading into the seventh, Martin Maldonado led off with a single against Jonny Venters (5-4) and pinch-runner Eric Farris went to second when pinch-hitter Jean Segura drew a walk.
Jeff Bianchi then laid down a sacrifice bunt to move the runners up and Farris scored the tying tun on Venters' throwing error on the play.
Chad Durbin came in to pitch, walking pinch-hitter Jonathan Lucroy, and Aoki followed by lining a pitch off the base of the left-center field wall, scoring a pair.
Carlos Gomez completed the rally with a sacrifice fly as the Brewers took the opener of a three-game series.
Jose Veras (4-4) got the win in relief of rookie Wily Peralta, who shut the Braves down on one run over six innings in his second major-league start.
John Axford collected his 28th save for the Brewers.
Martin Prado went 3-for-4 for the National League wild card-leading Braves.
Swim Week Miami Beach 2014 [PHOTOS]