The score had been tied 1-1 since the fourth inning, but Inge changed that when he sent a pitch from Phil Hughes (9-8) crashing off the facing of the right-center field stands to lead off the eighth.
Yoenis Cespedes also went deep for the Athletics, who won for the 13th time in 15 contests and clinched their four-game series against the Yankees.
Jarrod Parker (7-4) handled New York on one run over eight innings, yielding five hits and a walk while striking out five.
Sean Doolittle registered his first major league save by striking out the side in the ninth for the A's.
Hughes ended up allowing a pair of runs on four hits in a 7 2/3-inning effort.
Raul Ibanez had the only RBI for American League East-leading New York.