Minor (11-10) scattered three hits, didn't allow a walk and struck out four in racking up his fifth straight winning decision.
Chad Durbin, Luis Avilan and Jonny Venters combined for 1 2/3 scoreless relief frames and Craig Kimbrel collected his 41st save with a 1-2-3 ninth for the Braves, who have clinched a National League wild-card berth.
The loss went to New York's Chris Young (4-9), who gave up only two runs on seven hits through six innings.
'Happy' fan videos make Pharrell cry [VIDEO]
Pregnant Mila Kunis wins 'Best Villain' at MTV Movie Awards