Juan Carlos Oviedo (1-0), the third Tampa Bay reliever, pitched the ninth and got the win.
John Lackey gave up five hits, including three by Evan Longoria, in seven-plus shutout innings for the Red Sox, who have lost eight in a row. Boston has scored just 16 runs during its longest skid since dropping the final eight games of the 2012 season.
Tampa Bay starter Chris Archer struck out 11, walked four and scattered four hits over six scoreless innings.
Red Sox manager John Farrell tweaked his lineup, which failed early on Friday to get a key hit as Boston couldn't score after getting two on with two outs in the first, second and fifth innings.
Brock Holt hit leadoff for the second time this season. Dustin Pedroia made his second appearance batting third, with regular third- and fourth-place hitters David Ortiz and Mike Napoli both moving down one spot.
Ortiz stopped an 0-for-17 slide with an opposite-field broken-bat ground single to left in the first.
Farrell said second-place hitter Xander Bogaerts could bat leadoff against left-handed starters.
Boston right fielder Shane Victorino left in the ninth with right hamstring tightness. He appeared to get hurt running to first base.
NOTES: Tampa Bay manager Joe Maddon talked with RHP Grant Balfour one day after the closer voiced displeasure with fans who booed him after a blown save Thursday. ''I emphasized the point that it's important he says what he wants to say, and then I gave him my take on everything beyond that,'' Maddon said. ''But in no way would I try to muzzle any of our guys.'' ... Red Sox SS Stephen Drew was optioned to Single-A Greenville to start a stretch of nine or 10 minor league games. He went 1 for 2 and played five innings of defense Friday. ... Red Sox 3B Will Middlebrooks (broken right index finger) had X-rays taken to see how the healing process is progressing ... Red Sox LHP Felix Doubront (strained left shoulder) is ready to start a light throwing program. ... Rays LHP David Price (4-4) and Boston RHP Jake Peavy (1-2) are Saturday's scheduled starters. ... Rays 3B coach Tom Foley wore Tampa Bay senior adviser Don Zimmer's uniform No. 66. Zimmer is recovering from heart surgery last month.