The contest was scoreless when rain halted Tuesday's play after 8 1/2 innings, and it didn't take long for the Reds to settle the issue once the game was resumed at Great American Ball Park.
The Reds' right-fielder then lined Aumont's first pitch solidly to right, chasing home Cozart with the game-winner.
Aroldis Chapman (2-0) notched the victory with his 1-2-3 top of the ninth Tuesday.
Reds starter Homer Bailey had struck out 10 while yielding just two hits over eight scoreless innings.
Phillies starter Kyle Kendrick allowed a pair of hits over seven innings, striking out four.
Philadelphia has dropped two straight games.