Gillaspie and Alex Rios both drove in two runs for the White Sox, who delivered a 13-hit attack that allowed them to split the two-game series.
Rios hit a solo homer in the ninth inning and Alejandro De Aza also had a solo blast in the first for Chicago.
The White Sox took a 5-1 lead into the eighth inning and then held on despite reliever Addison Reed giving up three hits in the final frame.
Jake Peavy (4-1) captured the win, allowing one run on three hits in 6 2/3. Jeremy Hefner (0-4) remained winless after surrendering four runs on eight hits in six innings.