Carlos Santana finished with a homer, two RBI and three runs scored, Yan Gomes stroked a three-run homer and Asdrubal Cabrera added three hits with three runs scored for the Indians, who have won two of three.
Cleveland starter Corey Kluber (4-3) struck out nine over seven innings of two-run ball.
Lonnie Chisenhall had a career-high five hits in the win.
Dustin McGowan (2-2) had a rough outing on the bump for the Blue Jays, allowing four runs on nine hits and two walks over four-plus innings.
Santana walked to lead off the second, then swiped second base for his second steal of the season. After a strikeout by Cabrera, Murphy knocked Santana in with a single to right.
Cleveland had a chance to add on in the third with the bases loaded, but Cabrera grounded into an inning-ending 3-6-3 double play.
Chisenhall's single in the fourth plated Murphy for a 2-0 lead. Santana then clobbered a two-run shot to right with no outs that doubled up the hosts, 4-0, in the fifth. Cabrera followed the Santana homer with a single to right that ended McGowan's night.
Kluber retired the first 13 batters he faced to open the game before Adam Lind slapped a double into left with one out in the fifth. Lawrie brought Lind around with a double two batters later to cut the deficit to 4-1.
Kevin Pillar struck out with men on first and second to end the inning for Toronto.
Melky Cabrera singled with one out in the sixth before Bautista cut into the deficit with a double to center to make it 4-2.
Asdrubal Cabrera led off the seventh with a double and Murphy knocked him in a batter later with a single to right. Mike Aviles made it a four-run game with his RBI single that knocked in Gomes later in the frame.
With runners on first and second and one out in the eighth, Asdrubal Cabrera's single plated Jose Ramirez and Murphy added a two-run double to stretch the margin to 9-2.
Gomes' three-run shot highlighted a six-run ninth inning as the Indians won going away.
Francisco's RBI double and Thole's RBI single in the ninth accounted for the final margin.