Jose Bautista hit his 11th home run of the season and rookie J.P. Arencibia also went deep for Toronto.
Adrian Gonzalez homered twice for the Red Sox, the 11th multi-homer game of his career, and David Ortiz added a solo shot.
Rajai Davis hit a one-out single off Matt Albers (0-1) in the 10th and stole second on a pitchout, sliding in just ahead of the tag. Davis stole third on the very next pitch, and the Red Sox brought their infield in. Cooper won it with a fly to center.
Carlos Villanueva (1-0) pitched a scoreless inning for the win.
Cooper entered in the sixth as a replacement for Blue Jays shortstop Yunel Escobar, who left after he was hit just above the left ankle by a pitch from Jon Lester. Escobar is day-to-day.
The Red Sox, who had their three-game winning streak snapped, were trying to reach the .500 mark for the first time this season.
Toronto took a 6-5 lead when Cooper connected off Daniel Bard to begin the eighth, but Gonzalez answered by opening the ninth with a solo drive to left off Frank Francisco, who had his first blown save.
The Blue Jays had lost three straight and seven of nine coming in, but jumped on Lester with three runs in the first, taking advantage as the lefty walked three of his first four batters. Lester's third base on balls, to Edwin Encarnacion, came with the bases loaded. Aaron Hill and Juan Rivera followed with RBI grounders.
Boston got one back on Carl Crawford's RBI single in the second and cut it to one on a leadoff homer by Ortiz in the fourth, his fifth.
Gonzalez gave the Red Sox a 4-3 lead with a two-run shot to left in the fifth, his fifth, but Bautista tied it with a leadoff shot in the bottom half.
Arencibia restored Toronto's lead when he opened the sixth with a drive to left, his fifth.
The Red Sox tied it in the eighth against Toronto lefty Marc Rzepczynski. Jed Lowrie doubled and was replaced by pinch-runner Jose Iglesias, who went to third on Crawford's grounder. Jarrod Saltalamacchia tied it with a two-out single.
Lester, who failed to extend his winning streak to five starts, allowed five runs and seven hits in 5 1/3 innings. He matched a career high with five walks and struck out five.
Toronto rookie Kyle Drabek allowed four runs and eight hits in five innings. He walked three and struck out five.
Notes
Jacoby Ellsbury's first-inning single extended his hitting streak to 19 games, while Crawford's RBI single in the second extended his hitting streak to 10 games. … Blue Jays 1B Adam Lind (sore back) missed his second straight start. Manager John Farrell said Lind is feeling better, and has not yet had an MRI or X-rays.