Johnny Gaudreau tied it 3-3 for Columbus at 12:44 of the third period, scoring his 14th goal with a wrist shot after a long outlet by Gavin Bayreuther.
"It was sad a little to give up that goal, but I believe in our team always," Kuznetsov said. "Eventually we get the goal in overtime, which is nice. Everybody can smile now."
Trevor van Riemsdyk scored twice, and Charlie Lindgren made 31 saves for the Capitals (27-20-6), who ended a three-game road losing streak in their last game until Feb. 11 at the Boston Bruins.
"We talked before the game how important these two points were going into the break, feeling good, a little positive vibes when we come back for the home stretch," van Riemsdyk said.
Joonas Korpisalo made 30 saves for the Blue Jackets (15-32-4), who have lost three straight. Columbus is off until Feb. 10 against the Toronto Maple Leafs.
"I thought it was a solid game by us and a few mistakes," Columbus coach Brad Larsen said. "There's some breakdowns, and it cost us there at the end."