Sanchez (1-0) scattered three hits and worked around three walks while striking out seven for the Tigers, placing 67 of his 110 pitches for strikes.
Phil Coke finished up with two scoreless relief frames for the save as Detroit won its second straight in New York to open the best-of-seven 2012 American League Championship Series.
The Tigers broke a scoreless deadlock in the seventh inning when Quintin Berry led off against Hiroki Kuroda (0-1) with a ground rule double, went to third on Miguel Cabrera's single and scored on a Delmon Young fielder's choice grounder.
The Tigers were helped by controversial call in the next frame when they scored two more runs.
Replays showed Omar Infante was tagged out at second base after Austin Jackson's two-out single, but umpire Jeff Nelson called him safe. The blown call opened the door to subsequent run-scoring singles by Avisail Garcia and Cabrera to ice the game.
Kuroda ended up yielding three runs on five hits over 7 2/3 frames for the Yankees, who now must travel to Detroit for the next three games, should the series go that far.