The Blue Jays were trailing 5-4 with two out in the ninth when Bautista blasted a 1-2 pitch from closer Addison Reed into the White Sox bullpen, extending the contest.
Then in the 10th, Davis led off with a single, stole second, went to third on Adam's Lind's fly-out and came home when Ramon Troncoso (0-1) uncorked a wild pitch.
The Blue Jays added an insurance run on a Maicer Izturis double and an error by Chicago catcher Tyler Flowers, then used Casey Janssen's 13th save in the bottom of the 10th to close out the triumph and even their three-game series.
Conor Gillaspie bashed a three-run homer and Adam Dunn added a solo shot -- his third round-tripper in two games -- as the White Sox saw their three-game winning streak snapped.
2014: The Year in Music [PHOTOS]