LaRue's lone hit snapped a 2-2 tie with one out in the top of the sixth.
Nick Stavinoha drove in two runs with a first-inning groundout and a third-inning sacrifice fly, while Joe Thurston had two of the seven hits for the Cardinals, who won twice in the three-game series.
Cards pitchers have given up just nine runs in the last nine games.
Winner Todd Wellemeyer (5-4) surrendered two earned runs and five hits in 5 1/3 innings with one walk and five strikeouts. Ryan Franklin, the last of six pitchers, earned his 12th save.
Milwaukee scored its runs on one of two hits by Frank Catalanotto, a fourth-inning triple.
J.J. Hardy had three hits, while Catalanotto and Craig Counsell both had two hits for the Brewers, who have lost six of eight.
The losing pitcher was starter Manny Parra (3-6).