Rick Porcello (5.2-9-6-1-7, 98) gave up three home runs, including two in the second inning. Austin Meadows's three-run job down the right field line put the Rays up 5-0 and the Red Sox could do nothing more than tease fans with a seventh-inning rally.
Boston, in losing its last two games, has slid back to 10 games out.
The Red Sox trailed by three runs when they batted in the eighth. Sam Travis led off with a single, but Colin Poche struck out Christian Vázquez, Michael Chavis, and Jackie Bradley, all swinging.
Chaz Roe pitched the ninth for the Rays and gave up a leadoff single to Mookie Betts (3-for-5).