Eighth-inning doubles by Kevin Pillar and Jose Reyes produced the go-ahead run. Doubles by Anthony Gose and Brett Lawrie also played roles in runs. Jose Bautista opened the scoring with a first-inning home run.
Toronto has won four straight and 10 of its last 14 to move within a half-game of Baltimore and the Yankees for first-place in the AL East.
"We've got a quality team," said Bautista, who had two hits and reached base in all five plate appearances. "When we execute like we have the last couple of games, we do well.
"We have a lot of guys who have been around and have had success. We have to keep coming out and executing."
Right-hander Steve Delabar (3-0) retired four straight, including striking out the side in the eighth, to earn the victory.
The save by right-hander Casey Janssen was his third of the season and 19th straight in save situations going back to last year.
Bautista's homer was his 179th with Toronto, tying him with Jesse Barfield for fifth place in club history.
The Blue Jays broke a 1-1 tie in the seventh when Gose, a speedster called up Thursday from Triple-A Buffalo when Colby Rasmus went on the disabled list, led off by legging out a double to left, stole third and scored on Melky Cabrera's chopper to third.
"He wants to be here, and he's showing he belongs," Jays manager John Gibbons said of Gose. "Guys who do what he does constantly put pressure on the opponent."
Jays starter Mark Buehrle was denied his major league-leading eighth victory. He allowed two runs over 6 2-3 innings. His ERA rose to 2.11 and his road ERA went up to 1.00.
Buehrle had six assists, many of which had teammates telling him he threw harder to first base.
"I don't know if I say in my mind I know I've got to kind of let it loose," Buehrle said.
Pillar hit a cue shot down the right-field line off left-handed reliever Neal Cotts (1-3). Reyes' two-out dunker into right off Cotts scored Pillar.
Texas has lost four straight and six of its last seven to fall seven games behind first-place Oakland in the AL West.
Michael Choice's third-inning RBI single to left tied the score and ended Texas' scoreless streak at 15 innings.
"I've been through struggles as a player, as a coach and as a manager," Rangers manager Ron Washington said. "I have to lead and come out positive every day."
Texas starter Robbie Ross lasted only 4 1-3 innings. In the Rangers' last 18 games, they have gotten three quality starts — all from Yu Darvish.
Rangers first baseman Prince Fielder sat out with a herniated disk in his neck, ending his consecutive games streak at 547. Fielder received a nerve-root injection on Saturday morning to alleviate pain that he has felt since last season and, according to Texas general manager Jon Daniels, causing weakness in his left arm. The Rangers hope Fielder can return to the lineup on Tuesday.
One of the 13 Rangers currently on the disabled list, left-hander Martin Perez, confirmed before the game that he will undergo Tommy John surgery on Monday.
NOTES: Recently retired Michael Young, a seven-time All-Star with the Rangers, was honoured before the game. ... With Toronto C Dioner Navarro returning from the bereavement list, RHP Neil Wagner was optioned to Triple-A Buffalo. ... RH reliever Justin Germano, designated for assignment by Texas on Wednesday, cleared waivers and is deciding whether to accept an outright assignment to Triple-A Round Rock.