Alvarez (1-0), called up from Class AAA last week to replace the injured Anibal Sanchez in the Tigers' rotation, yielded one run on three hits. The 24-year-old left-hander walked one and struck out seven.
Drew Smyly tossed two innings of shutout relief and Joaquin Benoit collected his third save as American League Central-leading Detroit logged a fourth straight win overall.
The score was tied 1-1 in the sixth when Miguel Cabrera led off the bottom of the sixth with a walk and took second on a singled by Prince Fielder. Don Kelly then golfed a three-run homer over the right field fence.
Justin Masterson (8-5) allowed four runs on six hits over seven innings for the Indians, who dropped a seventh straight.
Ryan Raburn smacked a solo homer for Cleveland.