Christian Vazquez flied out to shallow right and the runners held.
Bogaerts, who entered the game in the 10th at shortstop, singled to right to put the Red Sox (45-35) into the lead.
Glenn Sparkman replaced Loup to make his major league debut.
A double steal moved runners to second and third and Deven Marrero drove in two runs with a single to give him three RBIs.
Blue Jays starter Marco Estrada allowed two hits, seven walks and two runs in 4 1/3 innings.
Boston starter Doug Fister allowed three runs, four hits and three walks (one intentional) in five innings.
Bradley Jr. led off the sixth with a double on a fly ball that left fielder Ezequiel Carrera lost in the twilight. After Vazquez walked, the runners moved up on Tzu-Wei Lin's comebacker to the pitcher and Marrero tied the game with a single to left.
Tulowitzki tied the game with his fifth homer of the season on the second pitch of the bottom of the sixth by Boston reliever Heath Hembree.
Ramirez doubled to center against Roberto Osuna with one out in the ninth, but the Blue Jays closer ended the inning on a pop out to third and a flyout to left.
Boyer pitched around a one-out single in the bottom of the ninth to send the game to extra innings. Loup survived three walks (two intentional) and a wild pitch in the top of the 10th and Boyer had a perfect bottom of the 10th.
The Blue Jays took a 3-0 lead in the first inning on the 22nd homer by Smoak.
Estrada, who had pitched around two walks in the first inning, survived walking the first three batters of the third. He retired Benintendi and Dustin Pedroia before striking out Mitch Moreland. Estrada issued his seventh walk of the game to Marrero to lead off the fifth inning and Betts followed with an infield single to third, the first Boston hit of the game. Pedroia scored both runners with a double to left-center field before Barnes replaced Estrada.
NOTES: Toronto RHP Glenn Sparkman (fractured right thumb) was reinstated from the 60-day disabled list Friday and RHP Chris Smith was optioned to Triple-A Buffalo. Sparkman was taken in the Rule 5 draft from the Kansas City Royals and had been on the DL all season. ... Boston 2B Dustin Pedroia returned to the lineup Friday after taking Thursday off to tend to some nagging injuries. ... Boston LHP Chris Sale (10-3, 2.77 ERA) will oppose Toronto LHP Francisco Liriano (4-3, 5.46 ERA) Saturday afternoon in the second game of the series.