Hunter connected against Bryan Shaw (1-2) to complete a comeback for the Twins, who couldn't hold an early 6-0 lead. Hunter also drove in two runs in the Twins' six-run third.
Trevor May (8-7) pitched a scoreless eighth, and Glen Perkins worked out of trouble in the ninth for his 30th save. Carlos Santana drew a one-out walk and pinch-hitter Ryan Raburn singled with two outs, but Jerry Sands struck out on three pitches to end the game.
The Twins were coming off a four-game sweep in Toronto that dropped them three games behind in the AL wild card race.
The Indians scored four runs in the third and five runs in the fourth. Yan Gomes had four RBIs, including a three-run homer in the fourth that gave Cleveland a 9-7 lead. Minnesota scored a run in the fifth, and Brian Dozier's leadoff homer in the sixth tied it 9-all.
Shaw retired Trevor Plouffe to start the ninth, but Hunter hit the first pitch into the right-field seats for his 18th homer.
Twins starter Mike Pelfrey couldn't hold the big lead and remained winless since June 7, a stretch of 10 starts. The right-hander allowed seven runs and 10 hits in 3 2-3 innings.
Michael Brantley had three RBIs for the Indians, including a two-run double in the fourth.
Cleveland starter Cody Anderson retired the first six hitters but didn't get through the third, allowing six runs in 2 2-3 innings.