Danks (1-0) corralled his initial victory in his fourth start since returning from the disabled list, yielding just one run on three hits and a walk. The left-hander struck at six.
Addison Reed fired a 1-2-3 ninth for his 18th save as the White Sox won for just the second time in 12 games.
Sean Doolittle (3-1) was tagged with the loss after Athletics starter Tommy Milone had limited the White Sox to one run on four hits over seven frames.
Josh Donaldson drove in the A's only run.