Minnesota kept pace with Chicago, staying within a game of the American League Central-leading White Sox with 17 games remaining in the season.
The first five batters produced a run and three hits off Slowey (12-9), but he shut down the Royals for the rest of his stint. Boof Bonser worked the eighth and Craig Breslow escaped the ninth without allowing a run despite giving up two hits.
Carlos Gomez singled in Jason Kubel in the second to make it a 1-1 game and Kubel put the Twins in front in the third with a ground-rule double that scored Alexi Casilla.
Kubel added an RBI single in the seventh.
The loss went to Kyle Davies (6-7), who gave up four runs on eight hits in six innings.