That brought up Simmons, who laced a pitch into left-center field, sending home a pair of runs.
Luis Avilan (3-0) earned the victory in relief of Julio Teheran, who had allowed only two hits over seven shutout innings.
Craig Kimbrel finished up with his 30th save for the National League East-leading Braves, who clinched their three-game series.
Joe Kelly tossed 6 1/3 scoreless frames for the NL Central-leading Cardinals.
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