Dosunmu also had 12 rebounds and 12 assists for only the third triple-double in school history.
Cockburn had 14 rebounds and recorded his eighth double-double in nine games for Illinois.
D'Mitrik Trice scored 19 points and Nate Reuvers added 11 for Wisconsin.
Illinois outrebounded the Badgers 46-18. The Illini shot 54% from the field to Wisconsin's 41%.