Minor (9-4), winless since June 9, yielded a pair of runs on six his over seven innings, issuing just one walk. The left-hander struck out seven.
The Braves were trailing 2-1 when Reed Johnson singled and Joey Terdoslavich doubled to lead off the fifth. Minor followed by pulling a 1-2 pitch down the third-base line, sending home both runners for the decisive runs.
Luis Avilan tossed a scoreless eighth and Craig Kimbrel notched his 26th save for National League East-leading Atlanta, which won its second in three games against the Reds.
Cincinnati's Homer Bailey (5-8) coughed up four runs on 10 hits over six innings, shouldering the loss.
Shin-Soo Choo went 2-for-4 with an RBI for the Reds.