Dunn hit his 23rd home run of the season off Tommy Hunter (3-2), one inning after committing an error that contributed to two Orioles' runs.
Addison Reed (4-1) picked up the win with an inning of scoreless relief for the Chicago.
The White Sox went ahead 2-0 in the sixth when Alexei Ramirez led off with a double and then scored on a single by Alex Rios -- who took second on a throw to the plate, advanced to third on a fly out to center, and then scored on Dunn's single.
Baltimore tied it in the eighth when Brian Roberts led off with a walk, took third on a double by Nate McLouth and scored on an error by Dunn that also sent McLouth to third. McLouth came in on a sacrifice fly by Nick Markakis.