Kelly (6-3) improved to 6-0 in his eight starts for St. Louis, and in the process increased the Cardinals' lead in the division to 1 1/2 games over Pittsburgh and 4 1/2 ahead of the Reds.
Kelly eight hits over six innings. The only run he allowed was a solo home run by Shin-Soo Choo.
Reds ace Mat Latos (13-5) took the loss after surrendering four runs in six-plus innings.
Matt Holiday went 2-for-3 with an RBI for the Cardinals, who won their ninth game in 11 starts.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys