Minor (13-5) surrendered six hits and a walk while striking out two for the National League East-leading Braves, who had dropped the first three games between the two National League pennant contenders.
Craig Kimbrel recorded the final four out for his 41st save.
Andrelton Simmons cracked a solo homer while Jordan Schafer and Elliot Johnson each went 2-for-4 with an RBI for the Braves.
Lance Lynn (13-8) yielded four runs on nine hits in seven innings to take the loss.
Allen Craig went 3-for-4 with an RBI for the Cards, who remained tied with Pittsburgh for the NL Central lead.