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.