The Rays won for the 19th time in 22 contests and moved within a half game of Boston for the top spot in the American League East. Tampa Bay can claim the division lead on Thursday when the teams finish off their four-game series.
Price (5-5) improved to 4-1 since coming off the disabled list. He did not walk a batter and struck out four while going the distance for the fourth time in his six-year career.
The lone Boston run came on a homer by Mike Napoli in the seventh.
Felix Doubront (7-4) absorbed the loss, giving up three runs on six hits in 6 2/3. It was Boston's fifth loss in seven games.