After the game, New York announced that A-Rod will be placed on the 15-day DL with a strained right quadriceps -- leaving the team without two of its most important players.
"He's a hard guy to replace," manager Joe Girardi said. "We have to find a way to get it done."
Magglio Ordonez got Detroit going with a two-run single off struggling right-hander Phil Hughes, and the $139 million Tigers (12-15) won for the sixth time in eight games as they try to recover from an 0-7 start.
Granderson also doubled, walked twice and scored three times from the leadoff spot, while No. 2 batter Placido Polanco had four hits and two runs in a matchup of baseball's top-spending teams.
"It's nice to win here, without a doubt. They're a very good team. They always are," Rogers said. "I know they didn't have all their guys -- I don't care."
Robinson Cano hit a two-run homer for the Yankees, who stranded 13 runners in their return home after playing 18 of the previous 20 games on the road.
A three-time MVP, Rodriguez left the ballpark before the game for an MRI that showed a Grade 2 strain of his quadriceps. He originally sustained the injury April 20 in Baltimore and missed three games before coming back for a four-game series in Cleveland that ended Monday.
The slugger was removed for a pinch hitter in the eighth inning of Monday night's win over the Indians and said he felt a "pull" while running the bases.
Girardi said the team was surprised to learn the strain was so severe. Rodriguez was not available to reporters after Tuesday night's loss.
"Hopefully, it'll be no more than the 15 days," general manager Brian Cashman said. "But we've got to get it right."
The club also is waiting for additional opinions on Posada's ailing throwing shoulder before determining whether the All-Star catcher needs surgery.
Playing with their entire projected lineup for only the second time this season, the Tigers took advantage of an ineffective Hughes (0-4), who has a 9.00 ERA after six starts.
Rogers (2-3) gave the youngster a lesson in winning without sharp stuff, holding New York in check despite giving up six hits and walking four in six innings.
The 43-year-old left-hander was the oldest pitcher to start against the Yankees since Hall of Famer Nolan Ryan did so for Texas at age 45 on June 1, 1992, according to the Elias Sports Bureau. New York won that game 7-1.
Rogers, drafted by the Rangers four years before the 21-year-old Hughes was born, snapped a three-start losing streak and earned a regular-season win against the Yankees for the first time since Aug. 17, 1993, with Texas. Of course, he also beat them 6-0 in Game 3 of the 2006 AL division series with Detroit.
Rogers, who mostly struggled while pitching for New York from 1996-97, entered with a 6.45 regular-season ERA against the Yankees before notching his 212th career win.
"Feeling young is not in the equation anymore," Rogers said. They've had me for a while, so hopefully it'll be my turn for a while -- for whatever time is left."
Booed by the crowd of 49,194, Hughes allowed six runs and eight hits in 3 2/3 innings. He also crossed up new catcher Chris Stewart a couple of times, including on one of his two wild pitches. Now, the Yankees must decide whether to stay patient with Hughes and keep him in the rotation throughout his second big league season.
"It's something we'll have to talk about, and we'll talk to the player, too," Cashman said. "But right now, I don't think we'll talk about that until tomorrow at least."
Granderson homered to center leading off the third, his third home run since returning from a broken right hand last Wednesday.
After Polanco doubled and moved up on a wild pitch, another ex-Yankee hurt New York. Sheffield, now batting .164, drove a 2-2 delivery over the left-field fence, and Hughes was left shaking his head near the mound.
As Sheffield returned to the dugout, he pointed to his uncle, Dwight Gooden, the former Yankees and Mets pitcher who was sitting behind home plate.
"I've always lived my life trying to make him proud of me," Sheffield said. "He's been an inspiration."
Rodriguez is 10-for-19 with five homers against Rogers. ... Yankees CF Melky Cabrera was rested. ... RHP Ross Ohlendorf relieved Hughes and struck out five in 3 1/3 scoreless innings of one-hit ball.