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.
"We may have to do that come post-season," Gonzalez said. "It was perfect timing. He hasn't pitched in three days, with a day off tomorrow, it made a lot of sense to push him and he came through."
Kimbrel enjoyed the extra work. He also liked sitting the dugout for a change between the eighth and ninth innings.
"I found myself talking and joking around," he said. "Then I said, "Wait I've got three more outs to get. But I was able to get back out there and get in the zone."
Notes: Minor is 6-5 in 17 starts with over six days rest. ... The Cardinals will play their next 13 games NL Central rivals Cincinnati and Pittsburgh. ... Atlanta opens a nine-game homestand on Tuesday with a three-game series against Cleveland. LHP Alex Wood (2-2, 2.50) will face Danny Salazar (1-1, 3.52) in the opener. St. Louis plays host to Cincinnati on Monday in the start of a three-game series. Tyler Lyons (2-4, 5.09) takes on Mike Leake (11-5, 3.12). ... Atlanta is 7-7 since ending a 14-game winning streak on Aug. 10.