Grant Balfour (1-0) stranded runners on second and third to end the ninth for the Rays, and Troy Percival worked around a one-out double to Johnny Damon for his fifth save.
Stuck in a season-long slump, Teixeira hit a three-run double with two outs in the eighth to make it 3-all. But with one out in the 10th, he stranded a runner at third with a shallow fly ball and slammed his helmet several times in the dugout.
Evan Longoria, Akinori Iwamura and Dioner Navarro also drove in runs for the defending American League champs, who are on the first leg of a three-stop, seven-game road trip.
The Rays got help from their speedy, stealing three bases. Carl Crawford easily swiped one of them, joining Bert Campaneris, Rickey Henderson and Corey Patterson as the only players since 1954 to steal a base in nine straight games. Crawford is perfect in 20 tries this season.
It was perhaps fitting that after going the entire game without a home run, it would ultimately end on one. There have now been 39 homers hit through 12 games at the new Yankee Stadium, many of them over a right-field wall that measures just 314 feet down the line.
Only a few thousand fans were left to see it after a 28-minute rain delay sent most of an already meager crowd home midway through the eighth inning. The rain held off until the final out, when it began falling heavily once again.
The game was an entertaining pitcher's duel between most of the way, with the Rays scratching out a 3-0 lead off A.J. Burnett through six innings.
Rays starter Andy Sonnanstine was cruising before running into trouble in the eighth. Ramiro Peno and Jose Molina reached on hits, and after Damon drew a two-out walk, Teixeira roped a double off J.P. Howell as a light rain suddenly turned into a downpour that sent fans scurrying for the stadium's spacious concourse.
The Rays took the lead in the third, when B.J. Upton singled and stole second. He moved up on a fly ball and trotted home on Longoria's double.
Tampa Bay added two more in the sixth on Iwamura's sacrifice fly and Navarro's two-out single.
Sonnanstine set the Yankees down in order in the seventh inning, striking out Nick Swisher to end it, and the affable outfielder was ejected by plate umpire James Hoye after complaining about the called third strike.
Notes: Yankees 3B Alex Rodriguez (hip surgery) went 1 for 5 in an extended spring training game. He's expected to play six innings of an intrasquad game Thursday and could be back when the Yankees open a three-game series Friday at Baltimore. ... Henderson stole a base in at least nine straight games three different times.
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