Aybar's blast came in the seventh inning against Brian Bruney after shortstop Derek Jeter -- who had been 4-for-4 as a designated hitter -- dropped a popup for an error, making all three runs unearned.
Jeff Mathis walked, then Aybar reached the seats in the right-field corner after narrowly missing in the same area earlier in the at-bat.
Asked which bothered him more, the walk to Mathis or the blast by Aybar, Bruney deferred.
"I don't know," Bruney said. "They're both pretty equally bad. Not getting guys out, that'd be the difference [in the game]."
Morales' three-run homer off Joba Chamberlain was the big blow in a four-run fifth inning that brought the Angels roaring back.
Figgins' RBI triple into the right-field corner and Bobby Abreu's run-scoring single gave the Angels the lead in the sixth. Aybar opened the inning against Mark Melancon with a single.
Joe Saunders yielded five earned runs in five innings before turning it over to the bullpen, which limited the Yankees to a run in the final four innings with Brian Fuentes -- who struck out Jorge Posada to leave two runners stranded -- registering his Major League-best 25th save.
The win went to Jason Bulger (4-1), who pitched a scoreless sixth. Melancon (0-1) took the loss.
"We need to get a better handle on our starting pitching," Angels manager Mike Scioscia said. "The only option is to come back offensively, and we're doing it. We swung the bats a notch above."
The Angels were 5-for-14 with runners in scoring position, collecting a total of 13 hits against Chamberlain, Melancon and Bruney.
Melky Cabrera's two-out single got the Yanks to within a run in the seventh before Darren Oliver retired Hideki Matsui on a fly ball to deep right, handled by Abreu, a former Yankee, to leave the bases loaded.
Facing an Angels lineup that will be without Hunter (adductor strain in his right side) and Guerrero (muscle strain behind his left knee) for at least two weeks, Chamberlain breezed into the fifth inning with a 5-1 lead.
Figgins singled leading off and stole second, scoring on Abreu's single to left. After an error by Alex Rodriguez on Juan Rivera's grounder, Morales launched a booming homer to right-center, his 15th, to bring the Angels even at 5.
"Kendry just jumped on a hanging breaking ball," Scioscia said. "Same with Erick [in the seventh]. He got a little cutter spin in front of the plate. You won't see him miss one by a few feet and hit one in the same at-bat like that very often."
Rivera, who stumbled and fell crossing first base on A-Rod's error after getting entangled with ex-Angels first baseman Mark Teixeira, left two innings later with tightness in his left quad and right hamstring.
Replaced by Reggie Willits, Rivera -- the AL's most productive hitter since June 1 with 33 RBIs -- was listed as day-to-day.
Returning to the DH role, Mike Napoli had a double, a single and a walk in four plate appearances, and Gary Matthews Jr. drove in a run with his 1,000th career hit, a second-inning single.
"Everybody in our lineup has confidence they can play," Scioscia said. "Look at Gary Matthews, Nap getting more at-bats. Those guys have confidence at the plate. We have some depth.
"Right now, there are two big bats out of the lineup, but we have a lot of confidence we have guys who can produce."
Saunders, who has struggled since starting the season 5-1, made a few positive strides in Scioscia's judgment.
"I liked his arm speed and his stuff," the manager said. "I thought he pitched inside better. They're a patient offensive club. He used a lot of pitches to get through five innings. I think it was a step forward for Joe. Hopefully, he'll be able to fine-tune it."
Saunders yielded three first-inning runs and another in the second. Rodriguez had an RBI double in the first, and Nick Swisher slashed a two-run single. Johnny Damon's RBI single after the second of four Jeter hits had Saunders trailing, 4-0, in the second.
The Angels got that run back on singles by Rivera and Matthews around a Napoli walk in the second.
A-Rod unloaded his 15th homer, a solo blast to center, in the fifth.
That was the extent of the Bronx bashing, the Angels claiming the home run derby, 2-1, courtesy of Morales and Aybar.