Jeff Niemann, competing for a spot in the rotation, and Jeremy Hellickson, one of the Rays' set starters, blanked the Pirates into the third inning while Tampa Bay (5-1) built a quick 5-0 lead over the Bucs' top two starters.
Three of the four second-inning runs off Burnett, Pittsburgh's Opening Day starter, were unearned due to errors by right fielder Brad Hawpe, who misplayed Jose Molina's RBI single, and third baseman Pedro Alvarez, who dropped Sean Rodriguez's two-out wind-blown popup for a three-base error.
Luke Scott's RBI double off Rodriguez in the third made it 5-0 Rays.
Gerrit Cole, the top-ranked pitching prospect in the Pirates' organization, made his debut in a big league environment and allowed one hit and one run in two innings. The Rays scored on him without a hit in the fifth, which Desmond Jennings led off with a walk, then stole second and third before scoring on a sacrifice fly by Yunel Escobar.
The Pirates (2-2) got on the board in the sixth as Jose Tabata led off with a double against Alex Torres and scored on Matt Hague's two-out single.
Former No. 1 overall pick Tim Beckham doubled and tripled for the Rays.
Up next for the Rays: Matt Moore will start Thursday against the Tigers, making his first appearance of the spring and he will be followed by Roberto Hernandez, who is fighting for the No. 5 spot in the rotation. Joel Peralta, Cesar Ramos and Jamey Wright are scheduled to pitch, as well.