Lucroy finished with seven RBI in the rout, which allowed the Brewers to salvage one win in a three-game interleague series.
Lucroy's solo homer in first inning put Milwaukee ahead 2-1. He hit a two-run single during a six-run third inning that broke the game open, and then capped his day and Milwaukee's scoring with a seventh-inning grand slam.
Zack Greinke (5-1) turned in his seventh consecutive start without a loss, baffling Minnesota on one run over 6 2/3 innings, yielding five hits while walking two and striking out six.
Twins starter Jason Marquis (2-4) was torched for eight runs on eight hits before exiting after only 1 2/3 innings.
Joe Mauer went 4-for-5 for Minnesota, which had a four-game winning streak snapped.