Howie Kendrick's infield single broke a 1-1 tie and Torii Hunter sent pitcher Sabathia (1-3) for an early shower with a two-run double in the seventh.
Matt Palmer (2-0) retired 14 consecutive batters after giving up a run-scoring single to New York's Hideki Matsui in the first.
"It's not about the guy I was facing on the mound," the 30-year-old journeyman told reporters. "I've felt I could always do it, given the opportunity and the chance.
"It's competing, showing the coaches and staff I'd love to be here."
Palmer held the Yankees to three hits and one run over 6 1/3 innings.
"It's a matter of trusting what his stuff can do," Angels manager Mike Scioscia said of Palmer. "He understands his movement is where his real talent comes from."
Sabathia, who signed a $161 million, seven-year deal in the offseason, lasted 6 2/3 innings, giving up eight hits and five runs while striking out five batters.
"I'd given up the lead, and it's just frustrating," he said.
"You lose a tie game up until that point, and when we're not scoring runs, it's up to the pitcher to go out and keep it close into the eighth and ninth innings."
The Angels added another run in the seventh after Sabathia left on Mike Napoli's single and scored a solo run in the eighth and two more in the ninth.
New York, who had a four-game winning streak snapped, scored three runs in the bottom of the ninth but could not rally enough.
