Martin Havlat had a goal and two assists and Denis Arkhipov, Jeffrey Hamilton and Tuomo Ruutu each had a goal and an assist for the Blackhawks, who almost lost first-period leads of 4-0 and 5-1.
Duncan Keith had the first three-point night of his career with three assists - all in the opening period. Bryan Smolinski added a goal.
The Blue Jackets were booed off the ice after the opening period but received a standing ovation after coming back with three goals in the second period to cut the lead to 5-4.
It stayed that way until Kukkonen scored on a 2-on-1 pass from Havlat with 6:15 left.
Rick Nash, David Vyborny, Fredrik Modin and Alexander Svitov had the Columbus goals, with Vyborny and Sergei Fedorov each adding two assists. Nash and Modin each contributed an assist.
Chicago improved to 6-1-3 under coach Denis Savard, hired Nov. 27 to replace the fired Trent Yawney.
Columbus failed to get a point for the first time in seven games.