The Tigers were trailing their division rivals 2-0 entering the ninth and facing Indians closer Chris Perez (4-2), but stunned the Tribe with four runs.
After Andy Dirks drew a walk, Avila drilled a pitch to the opposite field that cleared the left-field wall for a clutch, three-run shot.
Al Alburquerque (2-2) earned the win after retiring the final batter in the eighth and Joaquin Benoit notched his third save as the Tigers extended their American League Central lead over second-place Cleveland to four games.
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