"We certainly showed some fight and a lot of character tonight," Texas manager Ron Washington said. "We got the monkey off our backs."
With the bullpen depleted, Burnett (1-1) relieved in the 14th, his first relief appearance since Oct. 13, 2004, with Florida at Philadelphia. Burnett, who started in Texas on Sunday, became Toronto's ninth pitcher of the game, tying a team record.
Gerald Laird greeted Burnett with a single and went to third on Catalanotto's one-out double just inside the first-base line.
With the infield in, Laird tried to score on Michael Young's grounder and was thrown out at the plate. Catalanotto moved to third on the play and scored one pitch later when Burnett threw a pitch through catcher Gregg Zaun's legs.
Catalanotto said third base coach Matt Walbeck warned him to look out for a ball in the dirt.
"Two seconds before it happened he said, 'Cat, be ready for a wild pitch,'" Catalanotto said. "I got an extra five feet and I was ready for it. Once it got by I knew I had a chance to score.
"[Burnett] throws a pretty nasty curveball so you've got to be ready at all times when he's on the mound," he added.
Zaun shouldered the blame for letting Burnett's pitch reach the backstop.
"It's frustrating," Zaun said. "It's one thing when the other team takes it to you and beats you. But when something as simple as a ball getting between my legs is possibly the difference in the ballgame, that's hard to swallow."
Josh Hamilton followed with an RBI double. Hamilton finished 4-for-7 and drove in two runs -- for the first time in seven games, he did not score.
Burnett said he'll be ready again in time for his scheduled start Friday against Detroit.
"I'm ready to go, no problem," he said.
Catalanotto also homered and Jason Botts hit a solo homer and two-run double for the Rangers. Milton Bradley drew five walks, two of them intentional, and extended his hitting streak to 11 games while going 2-for-3.
Few fans from the announced crowd of 15,686 remained when the final pitch was thrown at midnight, four hours and 53 minutes after the game began.
Laird, who caught all 14 innings, was icing down his legs in the clubhouse but still smiling.
"I was feeling good through nine and then, all of a sudden, I started getting a little tired in the 12th," Laird said. "It was fun, I enjoyed every minute of it out there."
Toronto rookie Jesse Carlson struck out three straight to escape a bases-loaded, no-out jam in the 11th.
The Blue Jays put runners at first and third with one out in the 10th, but Joe Inglett fouled off a suicide squeeze attempt before grounding into an inning-ending double play.
Trailing 5-2, Toronto tied it with three runs in the eighth against reliever Joaquin Benoit. Frank Thomas walked, pinch-hitter Matt Stairs singled and Lyle Overbay walked before Benoit issued a bases-loaded walk to Marco Scutaro.
Wes Littleton came on to face pinch-hitter Inglett, who grounded to first baseman Ben Broussard, who had replaced Botts for defensive purposes in the seventh. Broussard tried to get a force at second, but threw wildly into left field, allowing two runs to score and tying it at 5.
Texas has made eight errors in its past six games and leads the AL with 17.
"He just hit wrong and felt a little tightness in his back," Washington said. "Rather than send him back out there on that turf, we just decided to shut him down. We'll re-evaluate him tomorrow."
Rangers left-hander Kason Gabbard gave up two runs and seven hits in six innings.
"I don't even know how long I sat in the clubhouse for, but it was a good win," Gabbard said.
Toronto righty Jesse Litsch gave up four runs and nine hits in five-plus innings. He walked none and struck out five.
Rangers RHP Kevin Millwood, hit on the left shin by Torii Hunter's line drive Tuesday, is expected to be fine in time for his next scheduled start Sunday at Boston. ... OF Shannon Stewart (sore groin) got the day off for Toronto, but could be ready to play Thursday, manager John Gibbons said. ... A hard liner by Thomas ripped the glove off Littleton's hand in the ninth, but the reliever recovered to get the out, sending the game into extra innings. ... The Blue Jays returned RHP Randy Wells, a Rule 5 draft pick, to the Chicago Cubs.