Adam Jones and Steve Pearce also homered for the Orioles, who have won four straight and extended their AL East lead to nine games over second-place Toronto.
Baltimore posted its first sweep in Chicago over the White Sox since July 1995.
Cruz's homer was a two-run shot in the fourth off Hector Noesi (7-9) that gave Baltimore a 3-2 lead. Cruz is one ahead of White Sox rookie Jose Abreu and Miami's Giancarlo Stanton for the homer lead.
The Orioles have also hit 161 as a team this year, the most in the majors.
Wei-Yin Chen (13-4) went 7 1-3 innings and allowed three runs and six hits.
Zach Britton picked up his 27th save in 30 tries in front of a Chicago crowd that included a fair amount of noisy, orange-clad Orioles fans.
