Chychrun scored his 16th goal, the most among NHL defensemen, at 15:16 to make it 5-4.
"We stuck with it and got ourselves back and gave ourselves a chance to tie it up at the end," Chychrun said. "It's not the time to be negative and on a downer. It's crunch time, and we need to stick with it."
Kane scored into an empty net with nine seconds left for the 6-4 final.
"They tried to push in the end and they almost made it, so I'm glad we won at the end, we scored that empty-netter," Korenar said.
NOTES: Timo Meier got his 100th NHL assist on Couture's goal. … Karlsson had two points (one goal, one assist) in his previous eight games. … The Sharks were 0-for-4 on the power play and are 0-for-20 in their past eight games. … Arizona is 3-1-1 against San Jose this season. … Coyotes forward Phil Kessel's five-game goal streak against the Sharks ended. He scored seven goals in the first four games against them this season and two in the final game last season.