The big inning gave the Indians a 7-4 advantage and they survived single runs by the Twins in the eighth and ninth.
Russ Canzler and Shin-Soo Choo combined for six of Cleveland's 12 hits with Canzler contributing a two-run homer in the fourth and an RBI single in the fifth.
It was a 4-4 game heading into the seventh, but the Indians batted around in that inning.
Tyler Robertson (1-2) walked the first two batters in the eighth before being relieved by Alex Burnett.
With runners on first and third and one away, Brantley gave the Indians the lead with a single to right. Canzler then reached on an infield single to load the bases, after which Casey Kotchman drove in a run with another infield single and Cord Phelps picked up an RBI by drawing a bases-loaded walk.