Minor had allowed only two hits through 5 2-3 innings -- a ground-rule double to Aoki and a single by Martin Maldonado -- and looked to be in line for his second victory over Milwaukee this season.
He struck out eight and walked two. After striking out Ryan Braun with his career-high 115th pitch for the second out in the sixth, Braves manager Fredi Gonzalez brought in Cory Gearrin. After Aramis Ramirez walked, Gearrin struck out Rickie Weeks swinging.
"It seems like there's always on guy on a staff who doesn't get run support," Gonzalez said. "And, that's Mike. He really battled, but had to throw 115 pitches."
The Brewers sent 10 batters to the plate in the seventh, but did all the damage on two hits and a sacrifice fly in snapping the Braves' five-game winning streak.
"Once Mike came out, it got ugly," Chipper Jones said. "We forced them to make perfect throws under duress and they did. Give them credit."
Martin Maldonado led off the seventh with a single off Jonny Venters (5-4), who was trying to protect a 1-0 lead for Minor. Pinch-hitter Jean Segura walked.
Milwaukee got a break when Jeff Bianchi bunted back to Venters, who fielded the sacrifice and threw wide of third baseman Chipper Jones. Pinch-runner Eric Farris scored on the throwing error and Segura and Bianchi each moved up a base.
"Mike threw the ball so well and my job is to come in and hold them down," Venter said. "I didn't do my job. I gave it up."
Venter said it was bad fundamentals.
"I felt like I got there quick, but I rushed it and made a bad throw."
Jones agreed that the error was costly.
"That play right there was the ballgame," he said.
Chad Durbin came on to face pinch-hitter Jonathan Lucroy and walked him to load the bases. Aoki then hit a 2-0 pitch to the wall in left for a two-run double. Lucroy scored on Carlos Gomez's sacrifice fly to complete the scoring.
Jose Veras (4-4) pitched the seventh and worked out of a bases-loaded jam to earn the victory.
John Axford worked the ninth for his 28th save.
Milwaukee's 16th win in 21 games has brought the Brewers back to the fringe of the NL wild-card race.
The Brewers pulled to five games back of St. Louis for the final NL wild-card spot. The Braves remained 5 1-2 games ahead of the Cardinals, 11-3 losers to San Diego, for home-field advantage in the one-game wild-card playoff.
The Braves had four runners thrown out on the base paths, committed two errors and issued seven walks. They also left seven men on base over the final six innings.
"A lot of good things happening without getting a lot of hits," Brewers manager Ron Roenicke said. "Great pitching. Nori, the throw to third. Gomey, the throw to third, Maldonado, the two throws, one on the steal, one on the pickoff. Some big plays defensively."
Wily Peralta, a September call-up, was looking for his second consecutive victory after beating Miami Wednesday in his first start. He allowed one run on seven hits in six innings with four strikeouts and a walk.
The Braves scored in the fifth when Dan Uggla singled to start the inning. Running on the pitch, he took third when Brian McCann's single slipped through the spot vacated by Bianchi as the Brewers shortstop broke to cover second. Uggla scored when Andrelton Simmons, who started his first game since he broke his pinky finger on July 8, hit into a double play.
Braun, who left Sunday's game against the St. Louis Cardinals in the sixth inning with a sore right wrist, was back in the lineup. He went 0 for 2 with two walks.
Jones received a rousing ovation when he came to bat in the first. Before he stepped into the batter's box, he looked toward the stands down each foul line and acknowledged the crowd's cheers by waving his batting helmet. His single in the fifth was his 2,715 career hit and tied him with Bill Buckner for 60th all-time.
NOTES: Prado recorded his 10th three-hit game of the season. Jason Heyward had his 38th multi-hit game. ... Braves RHP Ben Sheets, who is on the 15-day disabled list with right shoulder inflammation, threw approximately 20 pitches in two innings of simulated action. "I hope I get back on the mound," the right-hander said. "Really, tomorrow we'll see how I feel and go from there. I felt good today." Gonzalez said that Sheets, who is 4-4 with a 3.54 ERA in eight starts, could pitch out of the bullpen or even make an emergency start once he's cleared for action. "Any capacity they ask me to help in, I'd be willing to do," Sheets said. ... The Brewers plan to honor Jones before Wednesday's afternoon game.