The Reds jumped in front when Shin-Soo Choo scored after second baseman Johnny Mercer couldn't handle Joey Votto's grounder with none out, opening the door to four Cincinnati runs in the opening inning.
Devin Mesoraco capped the rally with a run-scoring double.
Votto drew a bases-loaded in the sixth for a 5-3 lead in what turned out to be the game-winning run as the Reds held on to clinch their three-game series in a contest delayed by rain for more than an hour in the first inning.
Mat Latos (9-3) registered the win despite yielding three runs (two earned) on four hits over five innings.
Aroldis Chapman stranded the winning run at third base in the ninth, fanning Mike McKenry and Mercer to end the game and notch his 23rd save.
Burnett (4-7) was touched for five runs (only two earned) on 10 hits over 5 2/3 innings.
Andrew McCutchen went 4-for-5 with a solo homer for Pittsburgh, which dropped a third straight.