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.
The Twins were coming off a four-game sweep in Toronto that dropped them three games behind in the AL wild card race.
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.