Garza (3-4) scattered six hits and two walks over his stellar outing to pick up his first win since May 5, snapping a personal four-start winless streak.
Lucroy finished 3-for-4 with two RBI and two runs scored, while Mark Reynolds slugged a two-run homer for the Brewers, who snapped a five-game slide against the Twins with their fifth win in six games overall.
Milwaukee will also host Tuesday's contest before traveling to Target Field for two more tilts to complete this four-game, home-and-home series.
Kyle Gibson (4-5) absorbed the loss after giving up four runs on six hits over six innings for the Twins, who took two of three from the New York Yankees over the weekend.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys