The Blue Jays were down to their final two outs in the ninth when Cooper belted a pitch from Ryan Cook over the right-center field wall, forging a 1-1 tie.
Facing Jerry Blevins (4-1) in the 11th, the A's had Edwin Encarnacion at second base and Rajai Davis on first with one out and Jeff Mathis at the plate when Toronto Manager John Farrell set the runners in motion.
Mathis swung and missed, striking out, but a throw to third by A's catcher George Kottaras sailed down the left field line, allowing Encarnacion to come home with the go-ahead run.
Moises Sierra followed with an RBI double for an insurance tally and Darren Oliver set Oakland down 1-2-3 in the bottom of the 11th for his second save, securing Toronto's victory.
Jesse Chavez (1-1) earned the win as the Blue Jays rebounded from a 15-inning loss to Oakland in Friday's contest.
Derek Norris notched the only RBI for the A's.