Bradley snapped an 0-for-21 slump with a game-tying double in the seventh, then tied it again with a two-run homer off Roberto Osuna (1-5) in the ninth. He had gone 1 for 32 before the two hits, and was hitless since a fourth-inning double against Tampa Bay on Sept. 12.
"It felt good," Bradley said. "It felt like I finally made solid contact."
Interim manager Torey Lovullo praised Bradley for putting his slump aside and producing.
"It showed signs of maturity that he can put that behind him despite the tough circumstances," Lovullo said. "I know it was a big day for him and he's very excited."
Twenty-four of Bradley's past 33 hits have gone for extra bases, including 15 doubles and seven home runs.
It was the first time in 74 such situations this season that the Blue Jays lost when leading after eight innings.
Xander Bogaerts also homered for the Red Sox, who are 9-9 against Toronto this season. The Blue Jays' lead in the AL East dropped to 3 1-2 games with the Yankees' 5-0 win over the Mets.
"Anytime we can win against the best right now, that's always a positive," Bradley said.
"I've got to do a better job," Osuna said. "I've got to locate my fastball better."
Bogaerts singled to right and Ortiz followed with an RBI single to left.
Bogaerts made it 6-4 when he scored on a wild pitch and Rusney Castillo added an RBI single to center, chasing Sanchez.
"We showed a lot of character," Lovullo said.
Toronto's Edwin Encarnacion went 2 for 3 with a homer and three RBI, including a go-ahead single in the eighth.
After striking out in his first two at-bats, Bogaerts opened the scoring with a leadoff homer off R.A. Dickey in the sixth, his sixth.
The Blue Jays answered in the bottom half when Encarnacion hit one off the facing of the second deck in left, his 34th.
Dickey left after Brock Holt's ground rule double to begin the seventh. Mark Lowe came on and got Blake Swihart to ground out, with Holt advancing to third. With the infield in, Bradley followed with a chopper off the glove of diving second baseman Cliff Pennington that ricocheted into shallow right.
"Anything would have helped," Bradley said when asked if the hit had boosted his confidence. "I'm just glad that I was able to get something going to help the team."
Toronto broke the tie with two runs off Noe Ramirez in the eighth. Josh Donaldson and Bautista both walked to begin the inning before Encarnacion delivered a broken-bat single through the left side, scoring Donaldson from second. Three batters later, Kevin Pillar hit an RBI single to right.
Dickey allowed two runs and six hits in six-plus innings. The knuckleballer, who is winless in three starts, remains stuck on 99 career wins.
Boston's Wade Miley allowed two runs and three hits in 6 2/3 innings.
CAN'T BEAT BOSTON
Dickey has faced the Red Sox six times this season without posting a win. He's 0-3 against the Red Sox. The last pitcher to go winless in six starts against Boston was Dick Fowler of the Philadelphia Athletics, who went 0-6 in 1946.
TAKING A TUMBLE
Before entering the game, RHP Jean Machi slipped and fell while descending the steps of the elevated bullpen bench. He suffered a contusion on his left ear.
Red Sox: 3B Pablo Sandoval (illness) sat for the second straight game.
Blue Jays: SS Troy Tulowitzki said he's "making strides" in his recovery from a broken left shoulder blade after initially being unable to lift his arm. There is no timetable for his return.
Red Sox: LHP Rich Hill (0-0, 0.00 ERA) is coming off his first major league start in more than six years, working seven scoreless innings against Tampa Bay.
Blue Jays: LHP Mark Buehrle (14-7, 3.66 ERA) allowed five runs and nine hits in 3 1/3 innings in a loss at Fenway Park on Sept. 7. He's 1-1 with a 5.23 ERA in two starts against Boston this season.
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