McAllister (3-3) scattered five hits, issued one walk and struck out four over 7 2/3 innings en route to the victory.
Rich Hill got the final out of the eighth inning and Chris Perez completed the whitewash with his fourth save as the Tribe logged a second straight win over the Athletics and seventh in eight games overall.
Yan Gomes provided the game's only run with a fifth-inning sacrifice fly following a error by A's second baseman Eric Sogard.
Oakland starter Tommy Milone (3-4) absorbed the hard-luck loss, allowing just the one unearned run on five hits over seven full frames. The left-hander didn't allow a walk and struck out five.
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