Los Angeles was trailing 5-1 in the seventh when it began the comeback. Chris Nelson and Grant Green each singled to lead off the inning against Rays starter Roberto Hernandez.
Jake McGee took the mound and promptly walked Peter Bourjos to load the bases, then surrendered an RBI groundout to J.B. Shuck and a two-run single to Aybar, lifting the Angels to within a run.
Then in the ninth, Rays closer Fernando Rodney (5-4) walked Green to lead off the inning. Bourjos forced him at second with a fielder's choice but stole second and moved to third when center fielder Desmond Jennings dropped Shuck's fly ball for an error.
That brought up Aybar, who lined a 1-1 pitch into the right-center field alley for a two-run double.
Ernesto Frieri (2-4) got the decision and Dane De La Rosa collected his second save as the Angels took the opener of a three-game series.
Jose Molina went 3-for-4 with three RBI for second-place Tampa Bay, which fell 2 1/2 games behind American League East-leading Boston.
The Rays, however, maintained a three-game lead in the race for an AL wild card spot.
Notable deaths of 2014 [PHOTOS]