Homer Bailey (9-6) struck out 10 without walking a batter over eight innings to get the win. He allowed six hits while recording his fourth triumph in as many starts.
Cozart gave the Reds the lead with his 10th homer of the season in the seventh. Bruce and Rolen both went deep in the seventh.
All of those blasts came off starter Marco Estrada (0-4), who is winless in 10 starts this season.
Norichika Aoki drove in Milwaukee's lone run with a double in the eighth.
Aroldis Chapman struck out two in a perfect ninth to pick up his 16th save.