Ike Davis drew a lead-off walk in the bottom of the 10th against Nate Jones (0-3) and went to second on a Juan Lagares sacrifice bunt.
Baxter then entered the game as a pinch-hitter and lined a 1-1 pitch into the right-field corner, giving the Mets a victory in the first contest of a two-game series.
Bobby Parnell (3-0) earned the victory in relief of starter Matt Harvey, who took a perfect game into the seventh inning.
The only runner the right-hander allowed over nine innings was a two-out infield single to Alex Rios in the seventh. Harvey struck out 12 in the 105-pitch performance.
He became one of just three pitchers in major league history to go at least nine innings with one or fewer hits and at least 12 strikeouts to end with a no-decision.
Chisox starter Hector Santiago tossed seven shutout frames, allowing four hits and two walks while fanning eight.