Griffin (12-9) yielded just one run on five hits and a walk in seven innings, striking out seven.
With the A's clinging to a 2-1 lead in the seventh, Stephen Vogt socked a solo homer for and insurance tally and Oakland put the contest away with two more runs in the eighth.
Coco Crisp also homered for second-place Oakland, which pulled to within one game of American League West-leading Texas with a sixth win seven contests overall.
Alex Torres (4-1) took the loss for the Rays after Jamey Wright lasted just 1 2/3 innings in his first start since 2007.
James Loney homered for Tampa Bay, which dropped a fourth straight and sixth in seven games.
The second-place Rays fell 5 1/2 games behind AL East-leading Boston but still hold a three-game lead over Baltimore in the AL wild card race.