Burnett (9-11) recorded his first 12-strikeout game since 2009 and ran his total for the year to 203 -- becoming the first pitcher in Pirates history to notch at least 200 strikeouts in a season.
The right-hander yielded four hits and three walks as Pittsburgh evened its three-game series with Cincinnati.
The second-place Bucs remained two games behind National League East leader St. Louis with seven games remaining, while the third-place Reds fell three games off the pace.
Cincinnati's Homer Bailey (11-11) was touched for four runs (two earned) on three hits in a 5 2/3-inning start.
Zack Cozart homered for the Reds, who came into the contest with a five-game lead over Washington for the second NL wild-card spot.