Alex Rodriguez homered for the Yankees, who failed to extend their eight-game road winning streak despite taking a 4-0 lead in the second.
Gary Matthews Jr. had an RBI single in the second for his 1,000th career hit, and Bobby Abreu drove in a run in the fifth to cut the Yankees' lead to 5-2. Morales then hit the first pitch he saw from Yankees starter Joba Chamberlain over the wall in center field to tie the game at 5.
Chone Figgins' RBI triple in the sixth gave Los Angeles a 6-5 lead.
Chamberlain, coming off one of his worst starts of the season where he gave up a career-high eight runs, allowed the run in the second before imploding in the fifth.
He pitched 4 1-3 innings and gave up five runs - four earned - and nine hits.
Mike Napoli reached on an atypical error by shortstop Derek Jeter in the seventh and Jeff Mathis was walked by Brian Bruney. With two outs, Aybar connected to deep right to make it 10-6.
Mark Melancon (0-1) took the loss for New York, with Jason Bulger (4-1) pitching an inning for the win.
Brian Fuentes struck out Jorge Posada with two runners on for his 25th save in 28 chances.
Angels starter Joe Saunders continued to struggle with his command, giving up the four early runs.
Jeter, who was 4 for 5, hit the first pitch for a leadoff single in the first and Johnny Damon was walked after starting the at-bat with an 0-2 count.