Price (3-4) didn't allow a walk in notching his fifth career complete game, striking out five in a 98-pitch effort.
It was the left-hander's first home victory in nearly a year, coming in his second start since returning from a six-week stint on the disabled list.
Chicago's John Danks (2-6) absorbed the loss, yielding three runs on five hits over seven full frames. He walked two and fanned five.
Josh Phegley smacked his first career homer for the White Sox.