Eloy Jimenez, Yasmani Grandal and Brian Goodwin also connected for the White Sox, who went 5-1 against the Cubs this year.
Robert, Jimenez and Goodwin each drove in three runs.
White Sox slugger Jose Abreu hit a two-run double in a seven-run fifth to get to 101 RBIs with 29 games left.
The reigning AL MVP drove in a career-high 123 runs in 2019. Frank Schwindel homered for the Cubs in the fourth.