The game had been tied since the second inning before the Indians put together the winning run against Rhiner Cruz (0-2).
Cleveland stayed in the thick of the American League wild card race, finishing the night a half game behind the front runners in that chase -- Texas and Tampa Bay.
Yan Gomes started the Cleveland 11th with a one-out single and moved to second on an infield hit by Asdrubal Cabrera.
Bryan Shaw (5-3) worked a perfect top of the 11th to record the win.
Both starting pitchers, Dallas Keuchel of Houston and Ubaldo Jimenez of Cleveland, allowed a run over seven innings.
Notable deaths of 2014 [PHOTOS]