The win marked the Athletics' seventh of their last 10 games, making them one of the hottest teams in the American League despite losing their previous two games.
They now improve to 11-10 on the season and sit half a game behind the Texas Rangers (11-9) for the lead in the AL West division.
Muncy hit 1-for-5 in the game with his lone RBI being the game-winning sac-fly. The third baseman is hitting .274 on the year with two home runs and six RBI.
Chicago, who had bases loaded with two outs in the bottom of the ninth when the game was tied 6-6, had bases once more in the 11th, but this time with no outs and couldn't get anything done again.
Athletics relief pitcher Jack Perkins got himself out of a jam after finishing the final 2.1 innings of work, earning his second win on the season.
His team was no better in the bottom of the 10th when they also had bases loaded and couldn't bring anyone in. But Muncy made sure not to let the game go on any longer.
Teammates Shea Langeliers, Jacob Wilson, and Nick Kurtz combined for six of the team's 10 hits, with the latter two hitting home runs.
The White Sox are now 3-7 in their last 10 games and fell to 7-14 on the season.