Burnett, who also won 12 games in 2002 and 2005, allowed one unearned run and six hits in seven innings. He walked four and struck out 10. It was the 20th time Burnett has struck out 10 or more in a game, and the fourth time this season.
"He was on tonight," Carl Crawford said of Burnett. "His slider was nice and sharp and his fastball was hard. He looked like he had his good stuff tonight."
Asked before the game whether the Blue Jays would deal Burnett by Thursday's trade deadline, general manager J.P. Ricciardi emphatically replied that the right-hander "is not going anywhere."
"I don't want to say never, but someone would really have to blow our socks off now," Ricciardi said. "Why would we trade him? We've worked too hard to get to this point to break up our team in August.
"If we have a good series here we're right back in this thing. Now we just have to go out and do it."
Burnett was pleased to hear he's likely staying in Toronto.
"We've got a good thing going here," Burnett said. "It's a big series for us right now. The way we're playing, we don't think we're out of it. Let's make a run at this thing.
"I've kind of felt that this is where I want to stay," Burnett added. "I've got a good relationship with everybody here and I've kind of felt that I wouldn't go anywhere. Maybe that's a big reason I'm pitching the way I'm pitching."
Burnett can opt out of the final two years of his five-year, $55-million contract after the season, but manager Cito Gaston is hoping he'll stick around.
"I'd be real happy if he was in spring training with us," Gaston said. "That's his choice."
Burnett has won six of eight starts for Toronto, which improved to 3-7 against Tampa Bay this season.
"I've always told everybody that when I'm healthy, I'm going to be good," Burnett said. "You saw toward the end of last season, I felt really good and I was healthy and my numbers showed."
Scott Downs worked the eighth and B.J. Ryan pitched a perfect ninth for his 20th save in 22 opportunities.
The Rays scored a single run for the second consecutive game, both losses. Tampa Bay is 2-9 on the road this month, but manager Joe Maddon remained upbeat.
"I really liked the way we played tonight," Maddon said. "The way we ran, the way we played defense, little things we did or tried to do. If we keep doing what we did tonight, even though it was a loss, I believe that's the kind of game that's going to get us into the playoffs.
Tampa Bay right-hander James Shields lost for just the second time in his past seven starts. Shields (9-7) allowed three runs and five hits in 7 1/3 innings. He struck out five without a walk.
"I didn't really feel too good before the game," Shields said. "I cut my bullpen short, didn't really feel good at all. Once I got out there I felt pretty good."
Shields cruised through the first four innings, allowing just one hit, but stumbled in the fifth. Rod Barajas led off with a single and Stairs followed with a two-run drive to center, his 11th.
"It was down and away at the knees," Shields said of the pitch. "He just went out and got it."
Shields set down 10 of the next 11 batters but was knocked out in the eighth, after a diving B.J. Upton couldn't reach Brad Wilkerson's sinking liner. The ball rolled to the wall, putting Wilkerson at third. Joe Inglett chased Shields with a triple over Upton's head.
Grant Balfour came on and induced Marco Scutaro to pop out before Alex Rios lined out sharply to right.
Sloppy defense by the Blue Jays helped Tampa Bay open the scoring in the third. With runners at first and second, Burnett fielded Crawford's bunt but threw high and wide to third, pulling Scott Rolen off the bag and allowing Akinori Iwamura to slide in safely. One out later, Iwamura scored when Lyle Overbay fumbled Carlos Pena's grounder.
Burnett kept Toronto's 2-1 lead intact in the seventh when Upton drew a one-out walk, stole second and advanced to third when Crawford reached on a wild-pitch third strike. Evan Longoria ended the inning by grounding into a 6-4-3 double play.
The Rays designated RHP Gary Glover for assignment Monday. Glover (left calf) was placed on the 15-day DL on July 20. The Rays hope Glover, 1-2 with a 5.86 ERA in 29 games, clears waivers and can be sent to Triple-A. "I think he's about ready to pitch well," Maddon said. ... Toronto purchased the contract of RHP Scott Richmond, placed LHP Brian Tallet (broken toe) on the 15-day DL and transferred 2B Aaron Hill from the 15-day to the 60-day DL.