Victor Martinez hit a two-run homer and Torii Hunter added a solo shot as the Tigers posted their sixth consecutive win and 15th in 18 games to raise their majors-best record to 27-12.
Ian Kinsler had four hits, and Cabrera added three and two RBIs as Detroit dominated in the first meeting since losing last season's AL championship series to the eventual World Series champion Red Sox in six games.
"It really comes down to the starting pitching and the bullpen," Detroit's first-year manager Brad Ausmus said. "The starters have set the tempo and got us deep into the game. We've also got some big hits. Miggy and Victor have been hitting well all season long."
Xander Bogaerts had two hits and an RBI for struggling Boston, which has lost a season-high four straight. It was the first time the Red Sox were swept at home in a multigame series by the Tigers since losing four in June of 1983.
"With the top of their order, any mistake we made, we paid for it," Boston manager John Farrell said. "They came in swinging the bats well, and when Jake (Peavy) missed his location, that lineup made us pay for it."
Boston's David Ortiz went 0-for-2 with two walks, ending his 10-game hitting streak.
Anibal Sanchez (1-2), activated from the 15-day disabled list earlier in the day after being sidelined since April 27 with a blister on his right middle finger, worked five innings for the victory. Last season's AL ERA champ gave up two runs -- one earned -- and five hits, striking out three and walking three, including one intentional to Ortiz on a 3-2 pitch.
"He was actually pretty crisp. Fastball, changeup, slider all looked pretty good," Ausmus said. "Command around the zone was pretty good. We wanted to stop him around 85ish pitches. He got us through five. A pretty good first start back off the DL."
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Trailing 1-0 in the third, the Tigers scored three runs off Peavy (1-2). Cabrera's RBI single tied it before Martinez followed with his homer into Boston's bullpen. They made it 4-1 on Cabrera's sacrifice fly in the fifth.
Peavy was tagged for five runs on 11 hits in six innings. He has given up at least one homer in each of his nine starts, the majors' longest active streak.
"We just caught a team when they were hot and they played better throughout the series than we did," Peavy said.
Hunter's homer, which cleared the Green Monster and left Fenway Park entirely, made it 6-2 in the seventh.
Boston loaded the bases with one out in the fifth, but managed only one run to cut it to 4-2 when Dustin Pedroia scored after third baseman Don Kelly threw low to the plate on a slow grounder for an error. Grady Sizemore then lined to Sanchez, who fired to third and doubled off Shane Victorino.
Mike Avila's RBI single increased Detroit's lead to 5-2 in the sixth.
The Red Sox -- held to one run in the first two games -- hadn't led in the series until Bogaerts' RBI single gave them a 1-0 edge in the second.
Peavy has thrown 1,999 career innings. ... To make room on the roster for Sanchez, Detroit optioned right-hander Justin Miller to Triple-A Toledo. He was 1-0 with a 3.48 ERA in seven relief appearances. ... The Tigers open the final portion of their nine-game road trip with three in Cleveland, starting Monday when LHP Drew Smyly (2-2, 2.70 ERA) faces RHP Corey Kluber (4-3, 3.38). ... Boston is off Monday before opening a three-game series at home against Toronto when LHP Felix Doubront (2-3, 4.54) goes against the Blue Jays' J.A. Happ (2-1, 3.57). After the series against Toronto, the Red Sox play 24 of their next 36 games away from Fenway. ... Peavy gave up a season-high six runs in his last start, Tuesday at Minnesota. ... The Tigers' rotation entered with a majors-best ERA of 2.66. ... The Tigers won a one-game "series" in Boston in 1984.
Copyright by STATS LLC and The Associated Press
BOSTON, MA - MAY 18: Anibal Sanchez #19 of the Detroit Tigers pitches against the Boston Red Sox in the first...
Miguel Cabrera Batting Average Since 2009 - 2 Outs, Runners in Scoring Position
*8-16, 2 HR this season
Victor Martinez has 10 home runs through 39 games this season, the fewest games to 10 home runs for him in a season in his career. Through his home run, V-Mart is hitting .393 with 6 HR in 61 at-bats this month.