The Indians have won two out of three in the four-game series, letting a late two-run lead get away before coming from behind in the ninth.
Aviles came on to run for Giambi and Carlos Santana was walked intentionally to load the bases.
Lonnie Chisenhall then hit a grounder to second on which Santana was erased while Brantley was scoring the tying run.
That brought up Drew Stubbs, who hit a ground ball to third base. An out was recorded at second, but Stubbs barely avoided the double play as Aviles was crossing the plate.
Joe Smith (4-0) gave up two runs in the eighth, his only inning of work, but still picked up the win. Vinnie Pestano retired the side in the ninth for his fifth save.
Baltimore lost for the fifth time in six games.
2014: The Year in Music [PHOTOS]