Price (17-6) yielded only one earned run on two hits in six innings, walking four and striking out two.
Evan Longoria hit a three-run homer, Ben Zobrist and Jason Bartlett added two-run bombs and Dan Johnson and B.J. Upton each connected for a solo shot as the second-place Rays to pulled to within 1 1/2 games of the New York Yankees in the American League East.
Boston starter Daisuke Matzusaka (9-5) was rocked for eight runs on eight hits in 4 2/3 frames.
Darnell McDonald homered and Victor Martinez drove in a pair of runs for the Red Sox.
Celebrity Families of 2014 [PHOTOS]