Happ (1-0) allowed a leadoff double in the first inning to Jacoby Ellsbury, but Boston went hitless the rest of the way. The Blue Jays left-hander pitched 5 1/3 scoreless innings, surrendering three walks while striking out six.
Steve Delabar, Aaron Loup and Sergio Santos teamed up for 3 2/3 frames of shutout relief, securing the Blue Jays' triumph as they evened their three-game set against the Red Sox.
After J.P. Arencibia's two-run blast gave Toronto a 2-0 advantage in the fourth, Rasmus blew the game open with a tremendous three-run shot in the sixth.
Boston starter Jon Lackey (0-1) exited with an injury after 4 1/3 innings, during which he yielded a pair of runs on five hits.
The team said the right-hander suffered a strained right biceps but it did not specify how severe the injury is. Lackey left the field in obvious discomfort while clutching his arm, ESPN.com reported.