The Blue Jackets were playing their first game without defenseman Zach Werenski, who sustained an upper-body injury in a collision with Anders Lee in a 2-0 loss at the New York Islanders on Saturday.
Clayton Keller gave the Coyotes a 1-0 lead on the power play with 49 seconds left in the first period.
He took a cross-ice pass from Christian Dvorak for an off-wing one-timer from the right circle that went off Columbus defenseman David Savard for his fifth goal and first in 10 games.
"It was at the end of the period and you don't want to get scored on, but it was a PK, it was a lucky bounce, so you can't really do anything against that," Kukan said.
Kukan, a defenseman, tied it 1-1 at 2:24 of the second. He scored his first regular-season NHL goal in his 63rd game with eight seconds left on the front end of double minor to Jakob Chychrun for high-sticking. Kukan has one goal in 10 Stanley Cup Playoff games.
"I think I had a good game," Kukan said. "I've got good confidence right now. I'm making plays and doing really good."
Christian Fischer gave the Coyotes a 2-1 lead at 9:35 with a shot off the left post on the rush.
"That's a team that thrives off the rush," Blue Jackets forward Nick Foligno said. "We talked about it before the game. All their opportunities, obviously the power-play one, were rush chances.
"You can't give a team, not freebies, but allow them to get to their game. Tonight, we did."
Lawson Crouse extended the lead to 3-1 at 16:39 when he redirected a shot by Soderberg.
"Sometimes when things aren't going well, you're not feeling good, sometimes you throw pucks to the net, it hits something and goes in," Coyotes coach Rick Tocchet said.
Texier cut it to 3-2 at 2:13 of the third with his fourth goal before Soderberg scored into an empty net at 19:25 for the 4-2 final.
"We played a good game from the start," Soderberg said. "We got the first goal. We were pretty solid over three periods."
