Jennings (3-for-5, three runs scored) connected on a solo shot in the second to put the Rays ahead 2-1 and went back-to-back with Matt Joyce in the fifth, capping the game's scoring.
Evan Longoria also went deep in Tampa Bay's victory, which broke a two-game skid.
Roberto Hernandez (4-6) collected the win, yielding three runs on seven hits over seven innings. He issued a pair of walks and struck out seven.
Boston starter Jon Lester (6-3) remained winless in five starts, surrendering seven runs on eight hits over 4 2/3 frames.
Mike Napoli drove in two runs for the American League East-leading Red Sox.
Notable deaths of 2014 [PHOTOS]