The Indians have won the opening two games of the three-game series and have defeated the Tigers five times in a row.
Brantley's first homer of the year, a three-run shot, came in the opening inning and Kotchman added a two-run blast in the fourth to give Cleveland a 5-1 lead.
The Indians added three runs in the fifth, two of them coming in on a single by Johnny Damon, giving Cleveland enough insurance to withstand a four-run fifth by the Tigers.
Jeanmar Gomez (4-4) collected the victory despite giving up six runs on seven hits in five innings. Chris Perez protected the three-run lead in the ninth for his 19th save, tops in the majors.
Max Scherzer (5-4) was roughed up for eight runs on seven hits in 4 1/3 innings to take the loss. The Tigers have dropped seven of their last nine games.