Mike Leake went the distance for the second time in his three-year career to get the victory and keep the latest run of Cincinnati success going.
The Reds won 14 out of 15 games, then lost five straight before putting together their current winning streak.
Dickey (15-4), a candidate for the Cy Young Award, lost for the third time in five starts. He gave up five runs on 10 hits in six innings and surrendered solo homers to Rolen and Frazier in the fourth to break up what had been a 1-1 tie.
Bruce added his home run in the sixth.
Leake (5-7) allowed four hits and a run while striking out four and walking no one. He threw 111 pitches.