Atlanta, which has the best record in the NL, salvaged the final game of the four-game set. Andrelton Simmons hit his 12th homer of the season for Braves, who finish a six-game road trip at 2-4.
St. Louis had a four-game winning streak snapped.
Minor may not claim to be the team ace - but he certainly performed like one on Sunday, halting a three-game losing streak.
He bounced back from the shortest outing of his career to give up one run on six hits. He struck out two and walked one. He was pitching on seven days rest after allowing four runs in 1 2-3 innings against Washington on Aug. 17.
"I felt like I was pounding the zone, hitting more spots," Minor said.
Minor did not allow a run until the sixth. He retired the side in order twice in the first five frames.
"He's developing into a big-time deal," Atlanta manager Fredi Gonzalez. "The Wainwrights of the world, the Kershaws of the world, they stop losing streaks and extend winning streaks.
He's becoming one of those guys."
Minor also beat the Cardinals 4-1 on July 26, starting the Braves on a 14-game winning streak.
"He was sharp today, he didn't give us a lot of mistakes to handle," Matheny said. "And he got out of any kind of jams that he got himself into."
Minor, who also legged out an infield hit in the seventh, recorded his team-high 19th quality start.
The win gave Minor a team-leading 13th win of the season ahead of Kris Medlen and Julio Teheran, who have 10 wins each.
Simmons paced the attack with a solo shot in the seventh that gave the Braves a 4-1 lead.
Closer Craig Kimbrel came on with two out in the eighth and picked up his 41st save in 44 opportunities. It was his first four-out save of the season. a trend that may continue in October.