Slowey (7-1) came into the game having received more run support per game (8.62) than any starting pitcher in the major leagues. His support on Wednesday, however, came from the Minnesota bull pen.
Jose Mijares, Matt Guerrier and Joe Nathan combined to allow one walk over the final three innings and that base runner was erased in a double play.
The Twins downed the Red Sox for the second straight game and dropped Boston into a tie with New York for first place in the American League East.
Minnesota took the lead with a three-run third, getting RBIs on a sacrifice fly by Joe Mauer, a double by Jason Kubel and a single by Brendan Harris.
Michael Cuddyer drove in the other Minnesota run when he was hit by a pitch with the bases loaded in the sixth.
Daisuke Matsuzaka (0-3) remained winless this season after spending much of the campaign on the disabled list. He surrendered three runs on nine hits in five innings.