Broxton, who robbed Jose Martinez of a homer in the second when he stretched his glove atop the center field wall, gave Milwaukee a 2-1 lead with a single in the fifth off reliever Brett Cecil (1-4).
Garza (5-5), making his first start since July 21 because of a right leg strain, gave up one earned run on four hits in 5 2/3 innings to give second-place Milwaukee its first home series win over St. Louis since July 2012.
Kolten Wong missed a tying home run by a few feet when his drive to right-center bounced off the wall in the eighth. He was stranded at second when Anthony Swarzak retired Tommy Pham on a groundout and Matt Carpenter on a flyout.
The Cardinals begin a weekend series in Cincinnati today. Mike Leake (7-9) will get the start for the cardinals, and the Reds will go with Asher Wojciechowski (1-1) on the mound. Game time is 5:30 with the pregame beginning at 4:45 on KTLO AM.
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