Swarzak (2-0) allowed just one run on two hits over five innings, and the Minnesota bullpen stymied the Cleveland bats in a 3-1 win Wednesday afternoon at Target Field.
Swarzak, whose last start came in Oct. 2012, struck out three and walked one, while the bullpen allowed just three hits over the final four frames, helping the Twins win two of three from Cleveland. The right-hander was making a spot start in place of the injured Kyle Gibson, who is suffering from tightness in his lower back.
Oswaldo Arcia added a solo homer and Danny Santana went 3-for-3 with two runs scored for Minnesota, which won for just the second time in six games since the All-Star break.
Jose Ramirez had two of the five hits and drove in the lone run for the Indians, who lost for just the third time in eight contests.
Cleveland starting pitcher Trevor Bauer (4-5) suffered the loss, yielding three runs on six hits, striking out seven and walking two, in six innings.
The team exchanged a pair of third-inning runs. Ramirez doubled home David Murphy to stake the Indians to a 1-0 lead in the top of the frame, but Sam Fuld singled home Santana to even the contest in the bottom of the inning.
Minnesota later went ahead for good in the fifth when Santana scored on a Brian Dozier double and added another run in the sixth on Arcia's solo homer in the sixth for a 3-1 edge.
Cleveland had a chance to tie the game in the eighth, but with runners on second and third and two outs Michael Brantley popped out to third base to end the inning.
Glen Perkins pitched the ninth for the Twins, earning his 24th save of the season.