Los Angeles rallied from an 8-1 deficit to tie the game in the sixth, then fell behind in the top of the eighth on Kyle Seager's RBI double.
The Angels, however, rallied in the bottom half of the inning to score twice and wound up leaving the bases loaded.
Aybar was thrown out at home later in the inning, but the Angels loaded the bases and scored the winning run when Albert Pujols came in on the walk to Callaspo.
Seager got Seattle started with a two-run homer in the first and the Mariners scored four times in the third to seemingly take control of the game.
Los Angeles battled back thanks chiefly to a five-run fifth that featured a solo homer by Bourjos and a three-run blast by Mark Trumbo.
The teams combined for 29 hits, 17 of them by the Angels.
Scott Downs (1-2) got credit for the win by recording the final out in the top of the eighth. Ernesto Frieri worked a perfect ninth for his 17th save.
Carter Capps (2-2) gave up the final Los Angeles run and was thus charged with the loss.