Lawrie's game winner was one of two Blue Jays homers to bounce out after striking the top of the wall. Just as costly were two Texas errors in Toronto's three-run seventh.
''They beat us,'' Rangers manager Ron Washington said. ''We may have helped them a little bit but they beat us.''
After Francisco Cordero blew a save opportunity in the top of the ninth, Lawrie won it in the bottom half with a leadoff homer against Mike Adams (0-1), a shot that bounced off the top of the wall in left center, his third.
''I knew he hit it well,'' Adams said. ''I didn't know if it was going to get out or not. I thought it might hit off the wall. Obviously it hit off the top of the wall and continued going.''
The same thing happened on Kelly Johnson's three run shot in the third, also to left center.
''I've never seen two of them,'' Washington said. ''Guess (the Blue Jays) know their ballpark pretty good.''
Jose Bautista also connected for Toronto, who rallied from 5-0 down to beat the team with the best record in baseball.
''It shows a lot about this team and the heart that we have,'' Lawrie said. ''We're not going to quit until the end.''
Cordero (1-1) got the win despite allowing Michael Young's game-tying single in the ninth.
Trailing 6-4 in the seventh inning, Toronto took advantage of some sloppy Texas fielding to score three runs and take the lead.
Leadoff hitter Eric Thames reached on a fielding error by shortstop Elvis Andrus, Lawrie singled and Colby Rasmus was safe when catcher Mike Napoli couldn't make a play on his slow roller in front of the plate, and was charged with an error.
Alexi Ogando came out of the bullpen to replace Robbie Ross and Jeff Mathis greeted him by driving in Thames on a fielder's choice, with Rasmus forced out at second, before Lawrie tied it at 6 by scoring on Yunel Escobar's grounder. Mathis stayed out of the double play by running on the pitch. Johnson followed with a single to center that Craig gentry bobbled, allowing Mathis to score the go-ahead run.
''We just didn't execute defensively,'' Washington said. ''It's the first time we haven't executed defensively and they took advantage of it.''
The rally completed Toronto's comeback from Texas' five-run, bat-around second.
Making his third career start, Blue Jays right-hander Drew Hutchison struck out the first four batters of the game before running into trouble.
David Murphy singled, Mike Napoli walked and Mitch Moreland singled to load the bases for Alberto Gonzalez, who hit an RBI single to right. Another run scored on Gentry's fielder's choice, with Gonzalez forced at second. Gentry stole second before Ian Kinsler lined a two-run double to right. Andrus capped the rally with an RBI single.
Toronto closed the deficit with a pair of homers in the third, Johnson drilled a three-run shot that bounced off the top of the wall in left center, his fifth. Bautista followed with a towering drive off the facing of the third deck, his fourth homer and first since April 23 at Kansas City.
Both homers came off Rangers starter Neftali Feliz, who allowed four runs and four hits in five-plus innings. He walked a career-high six and struck out four.
''The main thing was he just couldn't get the ball down,'' Washington said. ''He kept it elevated.''
Hutchison gave up six runs, five earned, and seven hits in six innings. He walked two and struck out five.
Rangers outfielder Josh Hamilton (back) sat for the second straight game and remains day to day, while third baseman Adrian Beltre was held out of the starting lineup but pinch hit for Gentry in the ninth, striking out.
Napoli left the game in the ninth complaining of vision problems in his right eye and was replaced by Yorvit Torrealba.
''I don't really know how to explain it,'' Napoli said. ''It was just something weird that was happening with my eye. It went away. I can see fine now. I'm going to go see a doctor.''
NOTES: The Blue Jays have blown six saves. ... With Hamilton out, Gentry made his second straight start in CF. Gonzalez replaced Beltre at 3B. ... Home plate umpire Sam Holbrook ejected Blue Jays infielder Omar Vizquel, 45, for arguing balls and strikes from the bench in the top of the fifth. ... Canadian-born Hall of Famer Ferguson Jenkins, who pitched for the Rangers for six seasons, attended the game. ... Rangers LH Matt Harrison faces Blue Jays LH Ricky Romero in Wednesday afternoon's finale. Game time is 12:37 EST.