Kluber (1-3) notched his initial triumph in his seventh major-league start for the Tribe, holding the Tigers to a pair of runs on six hits over six innings. The right-hander walked one and struck out four.
Cody Allen and Esmil Rogers each tossed one shutout relief inning and Vinnie Pestano finished up with his second save as the Indians claimed the first of a three-game set.
Asdrubal Cabrera drove in a pair of runs, including a sacrifice fly in the seventh inning to break a 2-2 tie off Darin Downs (1-1).
Delmon Young and Jhonny Peralta each had an RBI for Detroit, which came into the contest tied with the Chicago White Sox atop the American League Central.