The Indians won their fifth in a row, producing the tying run in the eighth and then coming up with the winning tally off Casey Fien (1-2) in the 10th. The teams combined for 24 hits.
Minnesota jumped in front 6-5 with a three-run seventh highlighted by Chris Parmelee's two-run homer.
Cleveland's tying run in the eighth came on a base-loaded ground out by Jason Kipnis that brought home Michael Brantley.
Aviles singled on a 2-0 pitch to open the Cleveland 10th and was sacrificed to second by pinch-hitter Ezequiel Carrera. Stubbs then came through with his big hit to make a winner out of Chris Perez (1-0).
Perez worked the top of the 10th as the fourth Cleveland pitcher and escaped a two-on, two-out jam when he retired Wilkin Ramirez on a fly ball to right.
Kipnis drove in four runs, two of them with a third-inning triple and another with a squeeze bunt in the sixth.