Damon, facing his former mates for the first time since leaving New York after last season as a free agent, connected on a solo shot in the fifth inning to give the Tigers a 4-2 lead.
Brennan Boesch went 2-for-3 with a triple and three runs batted in and Miguel Cabrera finished 2-for-3 for Detroit. Eddie Bonine (3-0) recorded the victory after tossing 2 1/3 scoreless innings in relief of starter Brad Thomas.
Thomas made an emergency start in place of Dontrelle Willis, who was scratched before the game with a high fever.
Mark Teixeira hit a two-run homer for the Yankees, who rallied from a 5-2 deficit to get within a run in eighth inning thanks to RBIs from Marcus Thames and Brett Gardner.
But Jose Valverde collected his ninth save for the Tigers by pitching a perfect ninth.
2014: The Year in Music [PHOTOS]