Slowey (8-5) surrendered only one run in six innings of work, scattering four hits, walking one and fanning four. Slowey's performance broke a string of three straight poor starts in which he had allowed 22 hits and 18 runs in 12 1/3 innings.
Brian Duensing, Matt Guerrier, Jose Mijares and Jon Rauch teamed up to toss three innings of scoreless relief for the Twins, who took the rubber match of a three-game series at Target Field and regained first place in the American League Central.
Morneau, whose sixth-inning solo shot gave Minnesota a 4-1 lead, added an RBI double and Jason Kubel got two hits and scored a run for the victors.
Tigers hurler Andy Oliver (0-2) committed a throwing error in the fifth that opened the door to a pair of Twins runs.