Cleveland starter T.J. House pitched seven scoreless innings, holding the Twins to four hits, during the second game.
With the game scoreless heading into the bottom of the fourth, Carlos Santana put the Indians on the scoreboard with a solo home run. It was his second of the afternoon. Santana was 2-for-4 with two RBIs in the second game.
It was the only run Ricky Nolasco (5-11) gave up in seven innings of work, but it still wasn't good enough for the win.
That was all the Cleveland pitching staff would need. House fanned eight Twins hitters, and Cleveland closer Chad Allen picked up his 20th save by working a perfect ninth.
In the first game, Kyle Gibson didn't have a good early afternoon on the mound and things didn't go much better at the plate either.
Gibson (11-11, 4.58 ERA) gave up seven runs in just three innings of work, getting pulled in favor of reliever Anthony Swarzak with the Twins in a 7-1 hole.
But Cleveland quickly responded, getting a two-run home run from 1st baseman Carlos Santana in the bottom of the first, and a solo shot from catcher Yan Gomes in the second, to go ahead 3-1. (Click here for an MLB.com video of the homer.)
Gomes and DH Michael Brantley continued to torment Twins pitchers the rest of the game.
Brantley finished with two hits, driving in two runs (both RBI singles to right field, one in the third and one in the sixth) and scoring twice. Gomes meanwhile knocked a two-run double to center field in the third inning to make it 6-1; the 27-year-old finished with a game high three RBI on two hits, with one run scored.
That four-run third inning was it for Gibson, who surrendered seven hits and seven earned runs. Swarzak threw two scoreless innings afterward, A.J. Achter allowed one run in his two innings of work, then Lester Oliveros finished it off.
Cleveland starter Corey Kluber (15-9, 2.45 ERA) kept the Twins off-balance for 8.1 innings. Minnesota mustered eight hits off 28-year-old righthander, but couldn't string anything together to get more than two runs off him, the final coming on a Trevor Plouffe single to left that scored Joe Mauer with one out in the top of the ninth.
Kluber finished with seven strikeouts, no walks and his third straight win.
Vargas and Oswaldo Arcia combined for four of the Twins' eight hits in the contest.