The Angels moved within 2 1/2 games of the second wild card berth in the American League, overcoming a 7-5 deficit over the final two innings.
Morales tied the game with his two-out homer after Mark Trumbo had singled. Los Angeles went on to load the bases later in the inning on two singles and a hit batsman, after which Torii Hunter drew a walk that forced in Mike Trout from third with the go-ahead run.
A fielder's choice grounder by Kole Calhoun in the ninth added an insurance run for the Angels.
Jordan Walden (2-2) got the win by throwing just two pitches while recording the final out of the seventh inning. Ernesto Frieri gave up a single in the ninth on his way to his 19th save.
Tim Collins (5-3), the second of the three pitchers Kansas City used in the eighth, gave up three hits without retiring a batter and took the loss.