Minter has pitched in a team-leading 44 games and is second on the Braves with 10 saves. The decision was made after Minter played catch on Friday.
Right-hander Nick Anderson was placed on the 60-day injured list Thursday because of a right shoulder strain.
The Braves entered Saturday 61-29 after a 9-0 win over the Chicago White Sox in a series opener. Atlanta boasts the best record in the majors despite already losing left-hander Dylan Lee (left shoulder inflammation) and right-hander Jesse Chavez (bruised left shin).
"We've used all the depth, that's for sure," manager Brian Snitker said. "Dylan Lee was a huge loss, what he did, that extra lefty, bridging innings in the sixth and seventh to try to get us to the matchup inning we liked for Minter and Nick. Those are the two guys that had been getting us to [close Raisel Iglesias]. Those are important guys and big holes in the bullpen."
Snitker said of Minter's injury: "I don't think [it is] real serious."
"Hopefully when he gets back, he'll be able to resume his normal workload," the manager said.
Atlanta recalled right-hander Dereck Rodríguez from Triple-A Gwinnett. He threw two scoreless innings in a previous appearance with Atlanta in June.