New York trailed 6-4 heading into the bottom of the sixth, but managed a run in that inning on a sacrifice fly, scored an unearned run in the eighth to tie it and produced the game-winner without the benefit of a hit in the ninth.
Jeter reached in the ninth on a one-out walk from Brayan Villarreal (0-1), moved to second on a walk to Curtis Granderson and went to third on a wild pitch. He scored from there when a 3-0 pitch to Alex Rodriguez from Villareal got away from catcher Alex Avila.
The pitch was in the dirt and bounced away from Avila after hitting a shin pad.
The ninth-inning run made a winner out of Mariano Rivera (1-1), who retired the side in the top half of the frame.
Rodriguez hit a solo homer for the Yankees in the fourth and Russell Martin had a two-run shot in the fifth.
Detroit starter Justin Verlander gave up four runs on seven hits in six innings without figuring in the decision.