Price (1-4) departed from the marquee pitching matchup because of left triceps tightness during Boston's eight-run third inning, which Drew finished with his towering homer off Jamey Wright.
Coming off a one-hit, complete game shutout of Toronto, Lester (6-0) allowed two runs in seven innings to help end the Rays' season-best six-game winning streak.
The Red Sox, meanwhile, stopped a three-game losing streak that's been part of a May slide. They are 5-9 this month after going a major league-best 18-8 in April.
Price left with one out in the third inning after giving up an RBI single to David Ortiz. The left-hander walked toward the plate after delivering the pitch to the Red Sox designated hitter and appeared to grimace.
Manager Joe Maddon, a team trainer and the entire Rays infield joined Price on the mound. Price walked off the field after a brief consultation and was replaced by Wright.
A 20-game winner last year, Price has struggled this season. Tampa Bay has lost seven of his nine starts, and Price uncharacteristically has given up eight leads over his last seven outings.
The Rays' ace has been bothered by allergies since before the start of spring training, but Price said this week he'd never use that as an excuse for his performance. He gave up two-out doubles to Dustin Pedroia in the first and Will Middlebrooks in the second without allowing further damage.
The Rays weren't as fortunate in the third, when Price walked Drew and gave up a single to Jacoby Ellsbury to begin Boston's biggest inning of the season.
Pedroia's RBI single gave the Red Sox a 1-0 lead. Ortiz's opposite-field single that just eluded diving third baseman Evan Longoria made it 2-0.
Mike Napoli doubled and Jonny Gomes singled off Wright to drive in the final two runs charged to Price, who allowed four runs and five hits in 2 1-3 innings. Jarrod Saltalamacchia walked with two outs to load the bases for Drew, who lifted a 1-1 pitch into the half-filled right-field stands at Tropicana Field.
Pedroia has a seven-game hitting streak, and he's had multiple hits in six of those games. Middlebrooks hit a solo homer off Kyle Farnsworth in the eighth.
Desmond Jennings tripled to drive in Tampa Bay's first run, then scored when Ben Zobrist grounded out.
NOTES: Price has not won at Tropicana Field since July 19, 2012, a stretch covering eight starts. ... Boston RHP Andrew Bailey (right biceps strain) felt fine one day after a bullpen session and will pitch in a simulated game Thursday. A decision on whether he will be activated from the 15-day disabled list or pitch in a minor league game will be made after the outing. ... Red Sox LHP Franklin Morales (lower back strain) is scheduled to make his fourth minor league appearance Saturday with Triple-A Pawtucket. ... Tampa Bay's Alex Cobb (4-2) will start Thursday night's series finale, his first outing since striking out 13 in 4 2-3 innings against San Diego to become the first pitcher in major league history to fan that many batters in an appearance lasting fewer than five innings. The Red Sox will counter with LHP Felix Doubront (3-1).