Martin Prado put Atlanta ahead to stay with a two-run homer, and Michael Bourn added a three-run homer in the ninth. Mike Minor (3-4) needed 103 pitches to get through five innings but allowed only one run, and four relievers completed a nine-hitter.
Giancarlo Stanton hit his 14th homer and Jose Reyes hit his first for the Marlins, but they went 0 for 13 with runners in scoring position and left 11 men on base. They totaled 16 hits in the series while being outscored 21-3, and were swept in their new ballpark for the first time.
The Braves, who have won six of their past seven games, earned their third road series sweep.
Mark Buehrle (5-6) allowed two runs and three hits in six innings. He took a one-hitter and a 1-0 lead into the sixth, but Bourn singled with two out and Prado hit a 2-2 pitch just inside the left-field foul pole.
Prado added an RBI single in the eighth.
Heyward homered onto the grass beyond the 418-foot sign with two out in the seventh.
Heyward connected again in the ninth for his second career multihomer game.
Atlanta’s four home runs were a season high.
Flashy glove work helped the Braves preserve a two-run lead in the seventh after a leadoff walk. Heyward made a sliding catch in right field to rob Omar Infante of a hit, and Andrelton Simmons followed with a diving backhand stop to start a 6-4-3 double play.
Stanton’s homer was originally ruled a double when the ball hit just inside the right-field foul pole and bounced back onto the field, but the call was reversed following a replay review.
The game was scoreless when Reyes homered leading off the fifth inning. The ball landed on top of the wall and bounced into the Marlins’ bullpen.
That was the only run allowed by Minor. He walked five, allowed four hits and stranded seven to win for the first time in his past eight starts.
Minor walked two in the first inning and loaded the bases with two out, but Logan Morrison struck out. Morrison went 0 for 4 and is batting .133 with runners in scoring position.
NOTES: Marlins OF Austin Kearns said his latest injury isn’t serious, and he’ll likely be activated this weekend. Kearns was already on the disabled list with a sore right hamstring when he was hit on the left elbow in a game Wednesday during a rehabilitation assignment. ... Bourn’s six homers are a career high. ... Marlins RHP Ricky Nolasco, scheduled to face the Rays on Friday, is 2-3 with an 8.51 ERA in five starts against them. ... The Braves begin a nine-game homestand Friday. They’ve played 23 home games, fewest in the NL.