Toronto's victory also spoiled any potential plans for a playoff celebration by Boston, which now has a magic number of one for claiming the American League Wild Card.
Rod Barajas, Jose Bautista and Aaron Hill each homered for Toronto (73-84), which snapped an eight-game losing streak at the Fens, dating back to Sept. 13, 2008.
"It's been real tough to win at this ballpark," said manager Cito Gaston, whose club matched a season high with its fourth consecutive victory. "I've always thought this was a tough place to manage. You never know when you have enough runs, because before you know it, it can get turned around on you the other way."
Toronto rapped out 14 hits to back a six-inning, four-run effort from Scott Richmond (8-10), who won for the first time in four career appearances against Boston.
A grateful Richmond thanked the Jays' bats for saving him on a night he tied a career with three home runs allowed.
"It was the offense tonight, plain and simple," Richmond said. "They were unbelievable. I was just trying to get them right back into the dugout as quick as I could. The guys did such a good job giving me a good cushion that I just tried to pound the strike zone and not mess around."
The Jays didn't mess around against Bowden (0-1), who struggled mightily in place of Beckett.
"If you're around the middle the plate, teams are going to do damage against you," infielder John McDonald said. "He probably didn't pitch as well as he wanted to tonight, but I'm sure he's going to do better the next time we face him."
Unfamiliarity with Bowden didn't seem to faze Toronto, which took full advantage of Beckett's absence.
"You're coming to the ballpark expecting to face Beckett," McDonald said. "You always want to play against the better players in the league, but I love the way the results happened today."
Hill jump-started the Jays' scoring barrage two batters into the game, hammering a 1-2 changeup from Bowden over the Green Monster for his 35th home run of the season. Edwin Encarnacion -- who complained of mild groin tightness after the game -- singled home Adam Lind three batters later, and Lyle Overbay followed with a two-run double to right-center field.
Kevin Youkilis halved Toronto's lead later in the first, unloading a towering two-run homer that settled into the third row of the Monster seats.
With two on and two outs in the third, Barajas slammed the first pitch he saw from Bowden over the left-field wall for a three-run dinger that stretched the Jays' advantage to 7-2.
Youkilis did his best to keep Boston (91-65) within striking distance in the bottom of the frame, ripping a solo shot to left for his second long ball of the night.
Bautista added to the home run party in the fourth, when he launched a prodigious two-run blast off Hunter Jones that cleared the Monster seats.
A 9-3 lead through four was plenty for Richmond, who settled down to yield just one run over the final three innings of his 102-pitch outing.
"It was fantastic just to let him go out there and relax, pitch his game, not worry about picking at corners as much and attack hitters," McDonald said. "It makes life a lot easier for our pitchers if we can give them a couple runs early."
Richmond did his best to reciprocate the offensive support.
"From the fourth to the sixth, I was like, 'These guys have done such a great job of getting me a good lead to work with,'" Richmond said. "I just tried to pound fastballs, execute, get ahead, work the edges once I got ahead and not try to nitpick too much."
A sacrifice fly from McDonald and Bautista's RBI single chased Jones in the fifth, arming the Jays with an 11-3 cushion after five.
The Red Sox plated runs in each of the next two innings, but their rally was halted with two on and none out in the seventh by a 62-minute rain delay before the game was officially called.