Magglio Ordonez had three hits - including a two-run double off Ian Kennedy - and scored twice for the Tigers, who got a strong effort from their bullpen and won for the eighth time in 10 games.
Bobby Abreu hit a three-run homer in the first inning, but failed to come through with the bases loaded in the sixth. The banged-up Yankees, who have lost six of eight, got more bad news before the game when they learned pitcher Phil Hughes is expected to be out until at least July with a stress fracture in one of his ribs.
With Alex Rodriguez and Jorge Posada also injured, Shelley Duncan batted cleanup for New York and hit an RBI single that tied it at 4 in the fifth.
With two on and two outs, Santiago ranged to his left for Melky Cabrera's grounder and made a difficult flip to second base for the inning-ending forceout.
Carlos Guillen singled leading off the sixth, Jacque Jones doubled and Santiago put Detroit ahead 6-4 with a two-run triple to right-center off Jonathan Albaladejo (0-1).
Batting ninth, Santiago filled in admirably for shortstop Edgar Renteria, rested by manager Jim Leyland after starting the first 28 games this year.
A steady drizzle fell through the middle innings, and Robertson tumbled headlong to the turf while breaking to cover first base on Robinson Cano's groundout to the right side in the sixth.
New York chased Robertson later in the inning and loaded the bases with two outs, but Zach Miner retired Abreu on a fly to center with a 3-1 pitch.
Ordonez singled in the seventh before Cabrera lined an opposite-field shot to right off Albaladejo for his sixth home run. Miner worked 2 1-3 scoreless innings and Francisco Cruceta finished Detroit's first sweep of at least three games at Yankee Stadium since April 1966.
Robertson allowed four runs and seven hits in 5 2-3 innings for his first win in five career outings in the Bronx. He also improved to 2-5 against the Yankees, beating them for the first time since July 2004.
The left-hander fell behind 3-0 after three batters. Johnny Damon walked, Derek Jeter singled and Abreu homered to right on a 3-1 pitch.
But the Tigers came back quickly against the 23-year-old Kennedy, winless in six outings this season. Santiago hit a bloop double to left, moved up on a wild pitch and scored on Curtis Granderson's groundout.
Placido Polanco doubled, Gary Sheffield walked and Ordonez drove a 1-2 delivery off the right-field fence for a two-run double that tied it at 3.
Cabrera followed with a triple that Abreu misplayed in deep right, giving Detroit a 4-3 lead.
