Atlanta's Scott Linebrink (3-2) took the mound to start the ninth and Stubbs pounced on the first pitch he threw for the decisive, opposite-field homer.
Francisco Cordero (4-3) was credited with the win thanks to a scoreless top of the ninth.
Dontrelle Willis lasted 6 1/3 innings for the Reds, allowing three runs on six hits.
Braves starter Brandon Beachy was victimized by Martin Prado's error prior to Phillips' two-homer homer in the first, yielding three runs (one earned) on five hits over six frames.
Brian McCann socked a long solo homer and Julio Lugo added a score-tying RBI single for the Braves, who finished a seven-game road trip with a 3-4 mark.
Swim Week Miami Beach 2014 [PHOTOS]