With Carolina on a power play and goalie Cam Ward off for an extra attacker, the Hurricanes' Tuomo Ruutu scored with 26.5 seconds left in regulation to tie at 2.
Chicago's Jonathan Toews and Ruutu traded goals to open the shootout. And then the Blackhawks' Patrick Kane and Carolina's Jussi Jokinen also scored to keep it even.
After both teams failed on their next attempts, Versteeg beat Ward to give the Blackhawks the advantage. Chicago goalie Cristobal Huet then stopped Sergei Samsonov as the Blackhawks raced to the ice to celebrate the win.
Versteeg and Troy Brouwer scored in regulation for Chicago.
Rod Brind'Amour also scored for Carolina. With the point for the regulation tie, Carolina tied Pittsburgh for sixth place in the Eastern Conference with 78 points, although the Hurricanes have one more loss than the Penguins.
Senators 3, Lightning 2, OT
At Ottawa, Mike Fisher beat Karri Ramo off a rebound at 2:38 of overtime to give Ottawa a win over Tampa Bay for its fourth straight victory.
Brian Elliott made 22 saves for his fourth victory in a row, and Ryan Shannon and Chris Kelly scored in regulation. Cory Murphy and Adam Hall scored for Tampa Bay, and Ramo stopped 24 shots. The Lightning have lost four straight and 15 of 19.
Ducks 4, Canucks 3, OT
At Anaheim, California, Scott Niedermayer scored at 3:37 of overtime as Anaheim beat Vancouver
Petteri Nokelainen, Teemu Selanne and Ryan Getzlaf also scored, while newcomers Erik Christensen, Ryan Whitney each had two assists, and Jonas Hiller made 32 saves to help the Ducks snap a three-game losing streak.
Anaheim had two other goals disallowed, one by Whitney because the puck went in through the side of the net, the other by Christensen because of a penalty against Ryan Getzlaf for goalie interference.
Alex Burrows, Kyle Wellwood and Ryan Kesler scored for Vancouver.