The Blue Jays were within one out of winning the game in the ninth, but Boston's Mike Napoli hit a two-run homer to force extra innings.
Boston then put two men on without scoring in the top of the 10th and the Red Sox did not get another chance.
Rajai Davis doubled to open the bottom half of the 10th against Brandon Workman (3-2) and Davis went to third on a ground out by Jose Bautusta.
Edwin Encarnacion was intentionally walked and then stole second, forcing the Red Sox to bring in their infield.
Lawrie hit a hot shot on an 0-2 pitch and Drew lunged to his left in an attempt to make a play. The ball caromed off his glove into center field, allowing Lawrie to easily score from third.
Toronto ended a three-game losing streak with the win going to Brad Lincoln (1-1), who worked the final 1 1/3 innings.
It was the fourth loss in six games for the Red Sox. Their lead over Tampa Bay in the American League East dwindled to three games.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys