Right-hander Steve Delabar (3-0) retired four straight, including striking out the side in the eighth, to earn the victory. The Blue Jays added an unearned run in the ninth.
Toronto has won three straight while Texas has lost four in a row.
Toronto took a 1-0 lead on a first-inning home run by Jose Bautista that was pulled down the left-field line with two outs.
Michael Choice's third-inning RBI single to left tied the score and ended Texas' scoreless streak at 15 innings.
Texas' Luis Sardinas greeted left-hander reliever Aaron Loop in the seventh with a single to left to tie the score 2-2.
The save by right-hander Casey Janssen was his third of the season and 19th straight in save situations going back to last year.
Jays starter Mark Buehrle, was denied his major league-leading eighth victory.
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.
Toronto moved two games over .500 for the first time since April 22.
The four-game losing streak equals the Rangers' season high.
Texas starter Robbie Ross lasted 4 1-3 innings. In the Rangers' last 18 games, they have gotten three quality starts - all from Yu Darvish.
Buehrle's overall ERA rose to 2.11 and his road ERA went up to 1.00.
Bautista's homer was his 179th with Toronto, tying him with Jesse Barfield for fifth place in club history.
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, caused weakness in his left arm. The Rangers hope Fielder can return to the lineup on Tuesday and won't have to become their 17th move to the disabled list this season.
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 honored 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.