Joe Inglett drove in the tiebreaking run with a sacrifice fly in the 10th inning, Ryan earned his first save in more than a year and the Blue Jays polished off their three-game sweep of the Rangers with a 5-4 victory.
The Blue Jays were swept by the Oakland Athletics in their previous series before arriving in Texas at 5 a.m. on Friday. They departed on Sunday for a two-game series in Baltimore in a far better frame of mind.
"It's huge," manager John Gibbons said. "We got swept by Oakland so we had to regroup. We had to come in here tired after a long trip. Our guys played some good baseball. We've got a good, solid ballclub and we're playing hard."
Toronto entered the series opener having lost 16 of 19 in Texas, but the Blue Jays completed their first sweep of three or more games in Texas since taking three from the Rangers Aug. 26-28, 1985.
Ryan was activated before the game barely 11 months after undergoing elbow ligament replacement surgery. His last save was April 11, 2007, and he had season-ending surgery less than a month later and began this season with Class-A Dunedin.
Ryan had to work for this one after giving up a leadoff triple to Marlon Byrd. But he retired David Murphy on a groundout, got Milton Bradley on a foul out, and retired Gerald Laird when Alex Rios made a running catch in right field.
"It's good to get the first one under your belt," said Ryan, who makes his offseason home in nearby Westlake. "Extra innings on the road, it means even more. That's why I worked hard all winter, to get the opportunity to come back. It wasn't the prettiest save but the bottom line is to make a pitch when you have to. We got a good outcome."
The Rangers left 10 on base as their offensive inconsistencies continue. Texas was expected to put runs on the board, but the Rangers are batting only .246 as a team.
"We had our chances," Murphy said. "It was a tough series. Our pitching kept us in the games but our offense isn't getting it done. We're not getting the back-to-back hits. We're frustrated right now."
Jesse Carlson (1-0) retired the final two batters in the ninth for his first major league victory.
With the game tied at 4, Frank Thomas was walked by Dustin Nippert (0-1) leading off the 10th, and after Marco Scutaro came in as a pinch-runner, Lyle Overbay also walked. Gregg Zaun's sacrifice moved the runners up, and Inglett's flyball to right scored Scutaro to give the Jays the lead.
"There's no better feeling than coming in and doing your job," Inglett said.
Texas threatened in the eighth, loading the bases with two outs when Murphy walked, Bradley delivered a pinch-hit single, and Laird walked. But Scott Downs struck out pinch-hitter Jason Botts on a called third strike to keep it tied at 4.
Rangers closer C.J. Wilson came into a tie game in the ninth, and after the Blue Jays put runners on first and second, Wilson struck out Rios and set down Vernon Wells on a fly ball to end the threat.
Texas' Scott Feldman, called up from Double-A Frisco before the game for his first big league start, allowed three runs and eight hits in six innings. Toronto starter A.J. Burnett gave up four runs and seven hits in 5 2-3 innings.
Texas' Frank Catalanotto, who started the day with an .087 average after a 2-for-23 start, went 2-for-3. ... Rangers president Nolan Ryan addressed game-day employees prior to the game, bringing a message of appreciation from the club for their efforts. ... Ryan said recent sloppy play by the team -- four errors and numerous mental mistakes in the previous two games -- has a negative effect on long-range marketing. "When you don't play well, it's disappointing," Ryan said. "When you lose, at least [the fans] feel good if you play well. You hate to see fans leaving in the sixth inning." ... Texas optioned RHP Kazuo Fukumori to Triple-A Oklahoma to open a roster spot for Feldman. ... Rios' streak of 22 games reaching base dating back to last season -- longest active in the league -- came to an end. ... It was the first extra-inning game of the season for both teams. ... Toronto's Rod Barajas left the game in the sixth inning with a groin injury.