The ace right-hander gave up four hits and improved to 5-0 in his last six starts.
The crowd booed loudly when manager Torey Lovullo walked to the mound to pull Greinke (6-2) after Jose Abreu's two-out double in the ninth.
Daniel Descalso hit a three-run homer and Paul Goldschmidt added a solo shot for the Diamondbacks during their sixth victory in seven games.
After totaling 24 runs in their last two games at Seattle over the weekend, the White Sox managed only Leury Garcia's solo homer against Greinke, who walked one in his third double-digit strikeout game this season.
