Minor (6-7) allowed one run on four hits and no walks in eight innings. His nine strikeouts matched his season high. He retired the last 13 batters he faced.
The Phillies' only run came on Chase Utley's homer in the fourth.
Kimbrel pitched a perfect ninth for his 30th save. Utley was caught looking at Kimbrel's 99-mph fastball for a strike to end the game.
The Braves are 6-2 against Philadelphia with five straight wins in the 2012 series.