Detroit pinch-hitter J.D. Martinez tied the game with a one-out solo shot off Cody Allen in the ninth, but Brantley laced a 3-2 pitch from Al Alburquerque (1-1) over the right-field wall in the 10th to send Cleveland to victory.
Jesus Aguilar picked up his first major league hit with an RBI single in the fifth and stroked a tiebreaking sacrifice fly in the seventh for the Indians, who had lost four straight before claiming the opener of this three-game set.
Scott Atchison (1-0) secured the win after spinning a perfect top of the 10th.
Detroit, which was coming off a three-game road sweep of the Red Sox, didn't get to Progressive Field until late Monday afternoon after mechanical problems with the team plane forced the Tigers to stay in Boston for an extra night.