Aviles' run-scoring fielder's choice put the Indians ahead 2-0 in the first inning and his three-run blast in the third broke the contest open.
The Cleveland second-baseman added a sacrifice fly and Carlos Santana went 4-for-5 with an RBI for the Tribe, who dropped Sunday's first contest 9-0.
Corey Kluber (2-0) yielded two runs on seven hits over seven innings the pick up the decision, striking out six without surrendering a walk.
Royals starter Will Smith (0-1), called up from Class AAA Omaha before the game, was pounded for six runs (four earned) on seven hits over four innings.
Chris Getz drove in a pair of runs for Kansas City.