In the 18th, Bonifacio collected a one-out single off Ross Wolf (1-1) and went to third on an errant pick-off throw with Josh Thole at the plate.
After Thole fouled out, Davis came through with a bouncing hit over the third base bag, sending home the game-winning run.
Aaron Loup (3-3), the Blue Jays' ninth pitcher, worked a scoreless top of the 18th for the victory after Brad Lincoln had tossed four shutout frames.
The Blue Jays had a 3-1 lead in the ninth but closer Casey Janssen blew his first save opportunity of the season when A.J. Pierzynski and Elvis Andrus each came through with an RBI to extend the contest.
Aaron Lind had four hits and Colby Rasmus added three, including a two-run triple, as Toronto clinched its three-game series.
Jeff Baker socked a solo homer for the Rangers.
Kim Kardashian, Kanye West reportedly set wedding date
Ray Liotta sues skin care company over use of likeness