Oliver Bjorkstrand scored for Columbus on the power play at 10:00 of the third period and at 16:00 to tie it 5-5.
"You're up 5-3 you want to finish off the game," Kane said. "They just kept coming, had some chances and ended up tying it up. Nice to get two points but would've been nice to give them nothing."
Alex DeBrincat scored the only shootout goal.
"Everyone wants to win, but one point's better than none," Blue Jackets forward Cam Atkinson said. "We'll look at the good things and work on it."
Carl Soderberg had a goal and two assists, and Brandon Hagel scored his first NHL goal for Chicago (10-6-4).
Patrik Laine had two power-play goals and an assist, Atkinson had a goal and an assist, and Seth Jones had two assists for Columbus (8-7-5).
"We have a lot of work to do, but at least we can point to some things that we took some steps in the right direction and continue to try to get better," Blue Jackets coach John Tortorella said.
Kevin Lankinen made 25 saves for the Blackhawks, including a penalty shot by Bjorkstrand at 3:36 of the second period to keep the game tied. He made five saves in overtime, including on Bjorkstrand in the final seconds.
"Without his saves at the end in overtime, we don't get to the shootout," Chicago coach Jeremy Colliton said. "He allowed us to stay alive long enough to come through and make the last play. I'm sure he wants to be better, but the fact is, he made the saves we needed."