The defending Stanley Cup champion Blackhawks began the evening with a three-point lead over both Dallas and Calgary for the final post-season berth in the Western Conference. Chicago moved within one point of seventh-place Anaheim.
Chicago took a 2-0 lead in the opening 6 minutes on goals by Patrick Kane and Brent Seabrook, but the Blue Jackets then ran off three straight scores. Samuel Pahlsson put Columbus in front by tipping in a shot with 6:38 to play in the third period.
Bryan Bickell tied it for Chicago 2:35 later by putting a wrist shot past Mathieu Garon.
Stalberg's shootout goal came in the third round.
Corey Crawford made 31 saves in regulation and overtime before turning aside all three Columbus attempts in the shootout. Garon stopped 26 shots.