Vogt drove in Matt Joyce and Jeff Keppinger with his second-inning shot down the right-field line, and Jose Molina added a run-scoring groundout in Tampa Bay's three-run frame.
Kuroda, 37, signed a one-year, $10 million contract with New York in January after previously pitching with the Dodgers, and is one of two pitchers that manager Joe Girardi has guaranteed a rotation spot to, along with Opening Day starter CC Sabathia. Kuroda allowed three hits in two innings, walking none and striking out two.
"The most important thing I wanted to accomplish was the feeling for the real game," Kuroda said through an interpreter. "I think I was able to do that, so I'm happy."
Rays right-hander James Shields had a perfect two-inning start, striking out Derek Jeter and Robinson Cano in his first frame. Right-handers Burke Badenhop and Josh Lueke each turned in two innings of scoreless ball to follow Shields.
"I felt good, felt really good," Shields said. "It felt kind of weird the first couple pitches, but that's just a normal first game of Spring Training. Other than that, it was good."
Rafael Soriano, beginning his second season with the Yankees after leading the American League in saves for the Rays in 2010, evaded trouble in a bumpy third inning. Soriano allowed a hit, a walk and a double steal before Cano's sharp defensive play on a Keppinger ground ball saved two runs.
Keppinger singled in the sixth inning for his second hit of the day, and later came around to score on a sacrifice fly to right field by Jesus Feliciano. The sac fly resulted in a double play after Vogt was caught trying to tag up from second to third on the play.
Up next: The Rays will split up Thursday, with one squad staying in Port Charlotte, Fla., to face the Tigers and the other heading to Fort Myers, Fla., to play the Twins. Both games are at 1:05 p.m. ET, and the Rays-Twins game can be heard on Gameday Audio. David Price will face the Tigers, making his second spring start, followed by relievers Fernando Rodney, Joel Peralta, Cesar Ramos and Jake McGee, as well as reigning American League Rookie of the Year Award winner Jeremy Hellickson. Right-hander Chris Archer will get the start on the road against Minnesota.