Pitching for the first time since straining the muscle below his right shoulder on June 28, Cueto (5-2) allowed five hits and struck out five with one walk for the win.
Cincinnati moved five games ahead of Washington for second NL wild card.
Cozart got things going for the Reds with his two-run shot to left field in the second inning. He added a two-RBI single in the fourth.
Cashner fires 1-hitter; Padres cool off Pirates
PITTSBURGH (AP) — Andrew Cashner carried a perfect game into the seventh inning and faced the minimum 27 batters in a one-hitter that sent San Diego Padres past the Pittsburgh Pirates 2-0 on Monday night.
Cashner (10-8) retired his first 18 batters before Jose Tabata grounded a clean single into right field leading off the seventh. Tabata was erased when Andrew McCutchen grounded into an inning-ending double play, and the right-hander cruised the rest of the way for his first complete game in 31 career starts.
Cashner struck out seven and did not walk a batter.
The Pirates lost for the second time in eight games after entering the day tied for first place in the NL Central with the St. Louis Cardinals, who played later at Colorado.