Lester pitched seven innings and allowed two runs while Boston's bats exploded for eight runs in the third.
"Just felt a little tightness in my triceps on a couple of fastballs," Price told reporters after being charged with the loss after giving up four runs in 2 1/3 innings.
Tampa Bay manager Joe Maddon said Price would be evaluated on Thursday and would likely miss his next start.
Stephen Drew blasted a grand slam in Boston's big third inning.
In other Major League Baseball games on Wednesday, the Seattle Mariners routed the New York Yankees 12-2, with Japan's Hisashi Iwakuma getting the win after working through seven innings to move to 5-1 on the season.
Raul Ibanez smashed a pair of home runs for the Mariners, including a grand slam in the first inning, and Kyle Seager added a three-run blast.
In the National League, the St Louis Cardinals had a 4-2 win over the New York Mets, their sixth victory in seven games.
St. Louis (26-13) broke a 2-2 tie when Daniel Descalso scored on a wild pitch in the seventh, and Yadier Molina added an RBI single in the eighth.
The Cardinals' division rivals Cincinnati (24-16) kept pace with a 4-0 shutout of Miami.
South Korean Choo Shin-soo hit a pair of home runs and finished 4-for-5 while starter Mike Leake combined with three relievers to keep the Marlins scoreless.
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