Alex Verdugo had three hits and two RBIs for the Red Sox, who were swept by the Rays in a four-game series in their first meeting at Tropicana Field in April.
“The beauty of putting the ball in play,” Boston manager Alex Cora said. “We took our walks, ran the bases well. It was a total team effort.”
Yandy Díaz hit a two-run double and Wander Franco had three hits for Tampa Bay (40-19), which lost for the fourth time in six games.
Boston shortstop Kiké Hernández made two throwing errors. The first one brought home a run.
Trailing 4-1, the Red Sox rallied against Jalen Beeks (2-3). Pablo Reyes hit an RBI single before Turner doubled off the Green Monster through a stiff breeze, making it 5-4.
“He got a changeup down in the zone," Cora said of Turner's hit. “Tough conditions, that was the only way you could get to the wall, to be a line drive, a hard line drive.”
Masataka Yoshida followed with a run-scoring double before Jarren Duran hit an RBI single off Cooper Criswell.
Playing the opener of a four-game series on an unseasonably cool day after Friday’s game was rained out and rescheduled for Monday, the Rays jumped ahead 3-0 in the second against Garrett Whitlock.
Díaz drove his two-run double into the left-field corner after Hernández’s throwing error allowed Harold Ramírez to score. Hernández made a nice sliding play on a grounder up the middle but fired wide of second from his knees trying to get a forceout.
Corey Kluber (3-6), recently demoted to the bullpen, worked a perfect inning for the victory. Kenley Jansen pitched the ninth for his 12th save, getting three strikeouts.
There were a handful of fly balls knocked down by a breeze gusting to 26 mph and blowing straight in. Red Sox left fielder Yoshida broke back on Franco’s fly before racing in with the ball falling for a double.
“Masa broke back on it, and you could see how much the wind was affecting stuff,” Turner said. “I guess I just kept mine low enough that wind didn't really affect it.”
Making his second start since spending a month on the injured list with ulnar neuritis of his throwing elbow, Whitlock gave up four runs and six hits in 4 2/3 innings.
It was Boston’s first scheduled doubleheader since Aug. 30, 1978.
YOU CAN’T CATCH ME
The Rays had their 26th straight stolen base when Franco got his 21st, extending the longest streak in team history.
Rays: Franco trotted into third slowly on a single and was checked by a trainer but remained in the game. ... Catcher Christian Bethancourt got up slowly favoring his left leg after Boston catcher Reese McGuire landed on him on a close play at the plate but stayed in. ... LHP Josh Fleming underwent an MRI for a sore left elbow.
Red Sox: Cora said INF Christian Arroyo (right hamstring) was set to return from his rehab assignment Saturday to be re-evaluated. “No guarantee he’ll be active here or go back and play more,” Cora said.
Rays RHP Tyler Glasnow (0-0, 6.23 ERA) is slated to start the nightcap. Cora said after the game that RHP Kutter Crawford (1-2, 3.30 ERA) would go for Boston.
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