Bailey (10-10) issued one walk and struck out eight, placing 69-of-106 pitches for strikes.
Manny Parra turned in a scoreless eighth and Aroldis Chapman struck out the side in the ninth for his 34th save as the third-place Reds clinched their three-game series.
They came into the contest trailing National League Central-leading Pittsburgh by 3 1/2 games and pulled to within 1 1/2 games of second-place St. Louis with the victory.
Todd Frazier provided the game's only run with a seventh-inning RBI double off Seth Maness (5-2).
Cards starter Michael Wacha hurled six scoreless innings, yielding three hits and two walks while fanning three.
Chipotle plans first price increase in 3 years
Justin Bieber crashes Drake Bell's album release party