Moustakas then came to the plate and belted a 2-1 pitch into the right-field corner, scoring both runners.
Salvador Perez followed with an RBI single for a 4-1 Kansas City lead.
Alcides Escobar went 2-for-4 with a homer and a run-scoring single for the Royals, who improved to 7-3 in September and kept their playoff hopes alive, moving within three game of Tampa Bay for the final American League wild card spot.
Jeremy Guthrie (14-10) got the win, allowing one run on nine hits over six innings. He didn't issue a walk and fanned two.
McAllister (7-9) was reached for four runs on six hits in a five-inning effort for the Indians, who remained 1 1/2 games in back of the Rays.
Michael Bourn drove in a pair of runs for the Tribe.