Phillips smashed a two-run homer in the third inning and added a run-scoring single in the ninth for the Reds, who won a post-season victory for the first time since 1995.
Jay Bruce added a long solo homer in the fourth as the NL Central champion Reds grabbed a road victory to begin the best-of-five NLDS.
Cincinnati starter Johnny Cueto came out of the game with back spasms after facing just one batter, and was replaced by Sam LeCure (1-0), who earned the win with 1 2/3 shutout relief frames.
Aroldis Chapman allowed a run in the ninth and loaded the bases with two outs but secured the win by striking out Buster Posey to end the game.
Matt Cain (0-1) took the loss for the NL West champion Giants, yielding three runs on five hits over five innings.
Posey homered for San Francisco.
Swim Week Miami Beach 2014 [PHOTOS]