J.B. Shuck sparked the winning rally with a leadoff single against Mike Montgomery (3-5), who was acquired in a deal with Seattle last week. Shuck advanced on Dioner Navarro's sacrifice and Saladino followed with a bouncer into center field.
Dan Jennings (4-2) got two outs for the win. The White Sox also got two victories in their last at-bat on Sunday against Detroit.
The Cubs trailed 4-2 before Dexter Fowler and Anthony Rizzo each hit an RBI single in the ninth. They had runners on first and second with two out when Jennings struck out Jason Heyward to escape the threat.