"It's tough to lose, you just hate losing games that you feel like you had a chance to win," said Peavy, who threw eight innings of two-run ball and walked away with a no-decision.
The White Sox have lost 16 of their past 19 games at Rogers Centre and fell to 5-7 in extra innings on the year after Cooper won a six-pitch at-bat with Jones by ripping a single to right field.
Jones entered after Leyson Septimo issued a one-out walk to Kelly Johnson in the 11th. After Edwin Encarnacion's single put runners on first and third, Cooper lined a ball right to Alex Rios, who caught it on a one-hop but had no chance to throw out Johnson at the plate. White Sox manager Robin Ventura said there was nothing Rios could have done differently on the play.
"Just get a ball to the outfield, stay out of the double play, get a ball deep enough to be able to drive the run in," said Cooper, when asked what he was thinking at the plate.
The game only went to extras thanks to Dunn, who changed everything with one swing of the bat.
Dunn, who had already blasted a fourth-inning homer that traveled 469 feet, led off the frame by going the
other way to tie the game at 2. It was Dunn's 33rd homer and 1,000th RBI of his career.
Other than Dunn's pair of homers, the White Sox couldn't muster any sort of offense and scored fewer than three runs for the third time in their past six games. Dunn had three of Chicago's seven hits on the night.
"Besides Adam, we didn't really get anything going," Ventura said. "Didn't look fluid or anything, so you hope it gets better. We have to get something else going besides waiting for home runs."
Before things got dramatic late, Peavy and Blue Jays starter Carlos Villanueva engaged in a classic pitchers' duel.
Peavy breezed through the first two innings but hit a pair of batters in the third which eventually helped the Blue Jays push one across the board for an early lead.
The White Sox didn't stay down for long, though.
Dunn's first homer of the game evened things up at 1 in the fourth. His towering blast was the first hit the White Sox mustered off Villanueva.
Both starters put an end to any offense after that until the seventh.
Peavy lost a battle with rookie Moises Sierra, who hit his first career homer and gave the Blue Jays a 2-1 lead.
Catcher Tyler Flowers called for a changeup down and away and said Peavy hit his spot. Both of them were fine with the location of the changeup, which was the first that Peavy had thrown to Sierra all night.
It was the first time that Peavy had faced Sierra and he admitted that it is difficult preparing for players he doesn't usually see.
"I felt OK. I just got ahead for the most part and threw a lot of first-pitch strikes," Peavy said. "It's a scrappy bunch. I know they got some guys hurt, some guys got an opportunity to play and obviously that Sierra kid came up big with a homer there."
The 31-year-old Peavy threw 86 of his 124 pitches for strikes, walked two and struck out six. Peavy, who was charged with his first no-decision since June 7, turned in his fourth consecutive quality start and lowered his ERA to 3.04.
Peavy has allowed two runs and struck out 13 over his last two starts against Toronto, a span of 15 1/3 innings.
"You know he is going to compete, there is no doubt about that," Flowers said about Peavy. "I expected it, just him warming up in the bullpen. He seemed to have a good feel for everything and a good feel for where he was throwing it."
Flowers also complimented Villanueva, who he said is a tough pitcher to figure out because of all the offspeed stuff he throws. Villanueva will throw his secondary stuff in any count and even threw a 61-mph curveball to Kevin Youkilis at one point.
Villanueva lasted a season-high seven innings and allowed five hits, one walk, while matching a career high with eight strikeouts.
A.J. Pierzynski extended his hitting streak to 14 games with a fourth-inning single.
The White Sox entered the game having won 10 of their last 13 one-run games and fell to 3-4 against the Blue Jays this season.