Couture tied it 4-4 at 4:14 of the third when he banked in a shot from a sharp angle off Binnington after Kevin Labanc, who had a goal and an assist, redirected a puck through the crease.
"I liked our response in the third," Couture said. "We came out with some energy."
But Kyrou made it 5-4 at 10:28 with a shot glove side during a delayed penalty. He used Sharks defenseman Nikolai Knyzhov as a screen after taking a drop pass from defenseman Colton Parayko
"[Parayko] made a good drop for me and I cut to the middle and tried to use the D as a screen there and it worked out," Kyrou said. "I was just happy in the moment and did my thing."
Labanc gave San Jose a 1-0 lead at 11:38 of the first period on a redirection of Burns' point shot, and Couture scored a power-play goal on a one-timer off a pass from Tomas Hertl at 14:18 to make it 2-0.
Faulk scored seven seconds after exiting the penalty box to cut it to 2-1 at 2:36 of the second period, using Mario Ferraro as a screen to beat Dubnyk with a wrist shot, and Hoffman tied it 2-2 at 4:09 with a redirection of Parayko's initial shot.
"It always feels good to get the first one out of the way, especially coming back here for our first game at home," Hoffman said. "Obviously, the fans aren't with us, but I thought it was a good team win and I'm glad that I was able to contribute to it."