Chicago increased its lead in the AL Central to three games.
Nate Jones (8-0) pitched 2 2/3 innings of one-hit relief and the White Sox won their fourth straight.
The game had been postponed by rain last Thursday and Monday's makeup was the final meeting of the season between the two front runners in the division.
Addison Reed, the last of three relievers in the ninth, got the final out for his 27th save in 31 chances.
Trailing 4-3, the White Sox loaded the bases for a third straight inning in the fifth, driving out Detroit starter Doug Fister (9-9).