The stretch is the second longest in team history, three short of a 12-game winning streak in June 2004.
"It's really impressive," Tampa Bay manager Joe Maddon said. "It's a total team concept going on right now."
John Lackey (11-7) came up short in his first attempt at his 150th win, giving up three runs and eight hits over seven innings.
Before the game, the Red Sox traded right-hander Jake Peavy and cash to San Francisco for minor league pitchers Edwin Escobar and Heath Hembree. Boston, last in the AL East one season after winning the World Series, has lost five straight.
"I have to say it's been a disappointing week. A little surprising even," general manager Ben Cherington said. "Anything we do between now and Thursday afternoon will be with a mind toward building as quickly as possible towards April of 2015."
The nonwaiver trade deadline is Thursday.
Zobrist made it 3-0 with his seventh-inning shot. Grant Balfour (1-3) went 1 2/3 innings for the win, and Jake McGee pitched the ninth for his 11th save.
The Rays (51-53) have used a 27-11 stretch since June 11 to go from also-rans to wild-card contenders.
"It was really nice to get a win, but at the end of the day it's a team win," Balfour said. "Right now that's what we're after. We went through a lot in the first two months, and we're trying to erase all those losses and turn them into wins."
Jeremy Hellickson, coming back from right elbow surgery and making his second start this season, gave up five hits over 4⅔ innings. Jeff Beliveau replaced Hellickson with a runner on and retired David Ortiz on a hard liner to first.
Desmond Jennings had a bunt single in the first inning and advanced to third on catcher Christian Vazquez's errant throw. Boston manager John Farrell argued with plate umpire Chad Fairchild about Jennings' route to first, but the call stood.
"We were told there was no deflection," Farrell said. "The ruling was he didn't interfere with the play."
Jennings scored on Joyce's single.
Kiermaier's run-scoring single in the fourth came right after Yunel Escobar was almost picked off second base, a close play Farrell decided not to challenge.
"A challenge would have been nice there," Lackey said. "He looked like he was pretty out. They still might not overturn it, but you never know. We've had it go both ways this year, but what are you going to do?"
Ortiz was given a double in the first after a fan caught his drive near the yellow home run line at the top of the right-field fence. The call stood after a boundary area video review. Ortiz was left stranded when Mike Napoli struck out.
Brock Holt was thrown out trying to score from second to complete an inning-ending double play in the third after Ortiz was retired on a grounder into shallow right field.
The Red Sox also failed to score after putting runners on first and third with none out in the sixth. Balfour got the final two outs of the inning.
"Consistency, that's been elusive," Farrell said. "The ability to string some base hits together has been elusive."
Boston went 0-for-7 with runners in scoring position and left nine on base in getting shut out for the 11th time this season.
RHP Allen Webster will come up from Triple-A Pawtucket to replace Peavy as Sunday's starter. The Rays' starter is RHP Chris Archer. ... Tampa Bay reliever Joel Peralta (illness) threw a scoreless inning for Class A Charlotte and could return Monday.
Copyright by STATS LLC and The Associated Press
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