Aybar homered with a man aboard in the fourth inning, putting the Angels up 5-1, and added a two-run triple to ignite a five-run sixth that turned the contest into a rout.
Andrew Romine went 3-for-4 with a run scored while J.B. Shuck, Hank Conger and Peter Bourjos each added two hits with an RBI as Los Angeles took the first of a four-game series.
Garrett Richards (5-6) overcame seven walks to earn the decision, allowing one run on two hits over five innings while striking out six.
Rays starter Chris Archer (8-6) was roughed up for five runs on nine hits in a 3 2/3-inning effort.
Ben Zobrist hit a two-run clout for Tampa Bay, which dropped a fifth straight and eighth in nine games.
The loss reduced the Rays' lead in the American League wild card race to two games over Baltimore and 2 1/2 over the New York Yankees.
Celebrity Families of 2014 [PHOTOS]