Encarnacion's RBI single in the first inning put the Blue Jays ahead to stay and he added a seventh-inning solo shot.
J.P. Arencibia provided a two-run single and Adam Lind went 2-for-4 with an RBI for Toronto, which evened its three-game set against the Red Sox.
Drabek (1-0), the 24-year-old son of 1990 Cy Young winner Doug Drabek, limited Boston to one run on three hits and three walks over 5 1/3 innings. He struck out four.
Red Sox starter Daniel Bard (1-0) yielded only one run over 5 1/3 frames, surrendering three hits and three walks.
Adrian Gonzalez drove in three runs for the Bosox, who lost for the fourth time in five games to start the season.
Swim Week Miami Beach 2014 [PHOTOS]