Miller (12-8) snapped a personal three-start winless streak, yielding just three hits in a 101-pitch performance. The right-hander didn't issue a walk and struck out six.
Matt Carpenter chipped in with a bases-empty home run as the Cards won a third straight against the Braves and their fourth consecutive game overall.
St. Louis came into the contest trailing National League Central-leading Pittsburgh by one game.
Atlanta's Julio Teheran (10-7) absorbed the loss, allowing four runs on five hits over six innings.
Freddie Freeman socked a solo homer for the NL East-leading Braves.