McAllister (4-1) stretched his undefeated streak to six starts, overcoming four walks while holding the Rays to one run. The right-hander yielded only three hits en route to a victory in the first of a four-game series.
Vinnie Pestano tossed 1 1/3 frames of scoreless relief and Chris Perez recorded his 26th save for the Indians, who broke a two-game skid.
Shin-Soo Choo and Jason Kipnis each doubled in a run in the third to give Cleveland a 3-0 lead.
Alex Cobb (4-6) lasted only 3 1/3 innings for the Rays, walking five batters while coughing up three runs (two earned) on five hits.
Jeff Keppinger went 2-for-3 with a solo homer for Tampa Bay.
Notable deaths of 2014 [PHOTOS]