The Blue Jays rallied for three runs in the seventh to grab a 7-6 lead -- an advantage they took into the top of the ninth, but Toronto closer Francisco Cordero (1-1) couldn't hold it.
The Rangers strung together consecutive two-out singles by Ian Kinsler, Elvis Andrus and Michael Young to forge a tie. Mike Adams (0-1) took over on the mound for Texas and Lawrie greeted him by blasting a 3-2 pitch off the top of the left field fence for the game-winner.
Kelly Johnson drove in four runs, including a three-run homer, as the Blue Jays evened their three-game series against the Rangers.
Kinsler finished with three RBI and Andrus had three hits for Texas.