Aybar hit a two-run double off closer Fernando Rodney (5-4) to put the Angels up 6-5 in the ninth.
Peter Bourjos stole second and went to third when center fielder Desmond Jennings misplayed J.B. Shuck's fly ball for a two-base error. Aybar then followed with the key hit.
Aybar also had a two-run single during a three-run seventh that made it 5-4.
Jose Molina had three hits and three RBIs for Tampa Bay, which has lost three in a row. The Rays, leading the AL wild-card race, are 2 games behind AL East-leading Boston.
After Ernesto Frieri (2-4) pitched a scoreless eighth, Dane De La Rosa got the final three outs for his second save.
Tampa Bay went up 3-1 during the fifth on Molina's third straight hit off C.J. Wilson, a two-run double. James Loney made it 5-1 with a two-run single in the sixth.
Molina entered 0 for 13 against Wilson.
Roberto Hernandez allowed three runs and eight hits in six innings. He left with two on and no outs in the seventh. Jake McGee entered and gave up a run-scoring grounder to Shuck and Aybar's two-run single that got the Angels within 5-4.
Wilson, who had gone 5-0 over his previous eight starts, gave up three runs, eight hits and five walks over five innings.
Mike Trout tied it at 1 in the fourth on his 23rd home run this season. It was his second homer in six at-bats against Hernandez, who struck out Trout with two on and two outs one inning later.
Molina put Tampa Bay up 1-0 on an RBI double in the second. Wilson avoided a big inning by getting an inning-ending bases-loaded fly ball from Ben Zobrist.
Tampa Bay's Wil Myers popped out with the bases loaded and two outs in the fourth.
NOTES: Tampa Bay optioned struggling RHP Jeremy Hellickson to Class A Charlotte for extra rest. He is not expected to pitch in the minors and should rejoin the Rays early next month. ... Angels INF Howie Kendrick (left knee) continues running and taking grounders. "We definitely expect him to play this year," manager Mike Scioscia said. ... Wilson tripped running out to pitch in the second and fell to the ground between the third base line and the mound.