The Yankees scored three times in both the third and fifth innings and were helped along by a costly error in left field by Coco Crisp.
New York has won three in a row and the Athletics have dropped three straight.
Curtis Granderson was on first with two away in the third inning when Cano hit a fly ball that Crisp misplayed for an error. Granderson scored on the play and Teixeira made the A's pay even more when he followed with a two-run homer.
Cano hit a solo shot in the fifth and Swisher had a two-run blast later in the inning to give New York a 6-1 lead.
Two of Oakland's runs came on solo homers from Josh Reddick in the fourth and Kila Ka'aihue in the seventh.
Ivan Nova (5-2) worked seven innings for the win, giving up six hits and striking out four.
Tyson Ross (2-5) was saddled with the loss. He gave up six runs, half of which were unearned, in 4 1/3 innings.