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 Pirates: Jameson Taillon makes his first appearance in a big league setting as he starts Thursday's game against the Red Sox at 1:05 p.m. ET at McKechnie Field. Jonathan Sanchez, who will work in relief, was originally scheduled to start but the two switched roles, possibly to better prepare Taillon for his upcoming start for Team Canada in the World Baseball Classic.