The score was tied 3-3 in the fifth when Young came up with one out against Indians starter Scott Kazmir and launched a 3-2 pitch over the left-field wall, putting the A's ahead to stay.
After Josh Reddick popped out, Callaspo also went deep, dropping a long fly ball just over the left-field fence for a two-run lead.
Young also provided an RBI single and made a great catch in center field, while Josh Donaldson drove in the three runs as the A's took 2-of-3 from Cleveland pulled to within a half-game of American League West-leading Texas.
Dan Otero (2-0) earned the decision with 1 2/3 innings of shutout relief.
Kazmir (7-6) was tagged for five runs on 10 hits over five innings for the Indians, who fell 4 1/2 games of out an AL wild card spot.