The Brewers scored five runs in the sixth inning to grab a 9-6 lead but the Blue Jays started a comeback in the seventh on RBI singles by Edwin Encarnacion and Yunel Escobar.
Then in the ninth, Rasmus connected for a game-tying solo blast into the corner of the right-field stands off Brewers closer John Axford (1-4), and Bautista followed with an opposite-field shot to right for the decisive blow.
Toronto belted six homers in the game, including two each from Rasmus and Bautista -- who now has 22 for the season -- while Encarnacion hit his 19th and Brett Lawrie also went deep in the win.
Aramis Ramirez belted a grand slam for the Brewers.
Notable deaths of 2014 [PHOTOS]