The 24-year-old Minor threw 26 pitches and loaded the bases in the first inning, but he escaped unscathed by striking out Logan Morrison looking.
"It was a tough first inning," Minor said. "I threw a lot of pitches, which caused me to only go five innings. 103 pitches, and that was a big part of it, but I felt like I battled the whole game."
Minor would settle in to throw four scoreless innings before Jose Reyes hit a ball that bounced off the top of the left-field wall and over for his first home run of the season. That was all the damage the Marlins would muster against Minor, who held Miami to one run on four hits over five innings to earn his first win since April 19.
"I thought he was right around the plate and just missed some pitches," Braves manager Fredi Gonzalez said. "He didn't buckle. There was progress there. He gave us five innings. There was a couple of innings where they could have put up a big number and he didn't let them do that."
Buehrle held the Braves to one hit over the first five innings, but the left-hander ran into trouble with two outs in the sixth. Michael Bourn singled, and then Martin Prado, who was 2-for-5 with three RBIs, put the Braves ahead with a two-run homer to left.
"That was big just because it was such a close game right there," Minor said of Prado's homer. "It just so happened to help me out to get the win."
Those were the only runs the Braves would score off Buehrle, collecting just three hits off the veteran lefty over six innings.
"The most frustrating thing is I made a good pitch on Prado," Buehrle said. "It was up and in. You just kind of tip your cap to him. ... We came out and were battling back and forth, 0-0, the whole time and we score a run and then that's the last thing we want to do, go out there and give up a run or two once it's a close game like that. But I made a good pitch. You just got to tip your cap to those guys."
Heyward added an insurance run in the seventh when he crushed a two-out homer to center off Steve Cishek. Prado added an RBI single in the eighth and Heyward went deep again in the ninth. Later in the ninth, Bourn broke the game open with a three-run homer.
"It helped out there," Heyward said of his first homer. "It was huge at the time, because it was still a close game and it gave us the extra run and then we added on later."
The Marlins appeared to threaten in the seventh, but the Braves' defense stepped up. Reyes led off with a walk, but Heyward robbed Omar Infante of extra bases with a diving catch in right, and Andrelton Simmons snared a Hanley Ramirez drive in the hole at short to turn a spectacular inning-ending double play.
"For me, the whole game was the seventh inning when Simmons turns the double play on Hanley and gets Jonny [Venters] out of that inning," Gonzalez said. "It kind of re-energized the team, and Heyward had two home runs and Michael Bourn hit that big shot in the ninth inning."
Giancarlo Stanton led off the eighth with a solo homer off Eric O'Flaherty. Initially ruled a double, umpires reviewed the replay and ruled the ball had hit beyond the right-field fence. Justin Ruggiano followed with a double, but O'Flaherty retired the next three batters to end the rally.
"We scored in the last four innings, and when they scored, we were able to get a shutdown inning," Gonzalez said. "We were able to add on, and one of the things you want to do defensively is when your team scores, you want to shut that other inning down. They weren't able to do it, and we were able to score some runs and add on."