With one out in the 11th, Jaso singled and moved to third on Gabe Kapler's single, which set up Bartlett's perfectly executed squeeze play. On the play, pitcher Jamey Wright (1-2) fielded the bunt but flipped the ball over catcher Mike Redmond's head as Jaso slid home with the winning run.
The play made a winner of Andy Sonnanstine (1-0), who had thrown a perfect top of the 11th.
Carl Crawford stroked a double and a triple and scored two runs in Tampa Bay's win, which put their MLB-best record at 27-11.
Hank Blalock also had a good night at the plate for the Rays, going 2-for-5 with an RBI.
Jhonny Peralta collected three hits for the Indians, who dropped their fifth game in seven contests.
Cleveland got more bad news when it learned that shortstop Asdrubal Cabrera sustained a broken arm in a collision with Peralta.