Edwin Encarnacion knocked in Bautista and later scored in the deciding sixth, and Toronto's bullpen did the rest behind Brandon Morrow, who labored through 3 2/3 innings but only allowed two runs on four hits and four walks.
Trevor Plouffe belted a solo homer while Chris Colabello recorded three of Minnesota's seven hits and knocked in a run for the Twins, who were coming off a three-game sweep of the Royals.
The Twins' early lead evaporated when Phil Hughes (0-1) imploded in the sixth.
Minnesota's starter survived a leadoff hit each of the first three innings and retired six in a row after giving up a double and a walk to open the fourth.
His luck, however, ran out in the sixth, which started with Munenori Kawasaki slapping a double down the third-base line. Bautista followed with a run- scoring single, and Encarnacion chased Hughes two batters later with an RBI base hit to center.
Michael Tonkin was unable to keep it a 2-2 game, as Dioner Navarro singled to load the bases for Moises Sierra, who pushed the go-ahead run across with a soft groundout to first. Lawrie brought in Encarnacion with a sacrifice fly that Kurt Suzuki compounded with a throwing error that allowed another run to score.
Hughes was charged with four runs on eight hits in five-plus innings of work.
The Twins had a chance to take him off the hook when they brought the tying run to the plate in the eighth, but Delabar induced an inning-ending force out off the bat of Brian Dozier.
Lawrie's grand slam in the ninth off Jared Burton, who loaded the bases with three consecutive two-out walks, provided some breathing room.
Plouffe took Santos deep in the bottom half to account for the final margin.
Minnesota scored a run in each of its first two at-bats. Dozier walked to open the home first and came around to score on Colabello's double off the wall in right, while Joe Mauer's single in the second frame plated Pedro Florimon, who reached base with two away when Bautista played his line drive to center field into a triple.