Alex Verdugo went 3-for-5 and drove in two runs while Turner, Masataka Yoshida and Pablo Reyes all had two hits apiece for Boston.
Corey Kluber (3-6) pitched a scoreless sixth inning to earn the win in relief.
Kenley Jansen worked around a hit to strike out the side for his 12th save.
Tampa Bay's Yandy Diaz and Wander Franco (3-for-5) each hit two doubles while going a combined 5-for-10.
Diaz (2-for-5) and Harold Ramirez (2-for-3) had two RBI apiece.
The Rays tagged Boston starter Garrett Whitlock for three runs with two outs in the second. After a Ramirez leadoff single and a two-out walk to Taylor Walls, Christian Bethancourt's ground ball single into the shortstop hole allowed Tampa Bay to draw first blood
Boston shortstop Enrique Hernandez's throwing error allowed Walls to take an extra base, and Diaz brought him home on a two-run double that dropped just fair inside the left-field line.
Ramirez added to the Tampa lead with a sacrifice fly in the fifth, ending Whitlock's day after 4 2/3 innings. He allowed four runs on six hits and struck out five.
Alex Verdugo lined a sharp run-scoring single off Diaz's glove at first, helping Boston get on the board with one out in the next half-inning. A single and a walk loaded the bases thereafter, but Jarren Duran struck out to keep it a 4-1 game.
The Red Sox flipped the game on its head with a six-run sixth, which began with Reyes knocking a two-on, one-out RBI single into right
After Verdugo struck out and Rafael Devers walked, Turner delivered the inning's biggest blow in the form of a three-run double high off the Green Monster. Yoshida followed with an RBI double of his own.
The teams traded runs in the seventh. Ramirez made it a two-run game again with an RBI single past the outstretched Hernandez, but Verdugo lined a two-out double to score Reyes, who singled, stole second and took third on a subsequent throwing error.
Tampa Bay opener Trevor Kelley pitched 2 ? innings of scoreless, one-hit ball with two strikeouts.
Jalen Beeks (2-3) took the loss, allowing six runs on four hits while recording just two outs.
--Field Level Media