Kuzmenko tied it 5-5 at 16:07 of the third period with a power-play goal when he redirected a Pettersson slap shot at the left post.
"I think a lot of times a team like San Jose, once they get the lead, they're starting to panic a little bit because they haven't had a lot of wins," Vancouver coach Bruce Boudreau said.
Dakota Joshua scored twice, Bo Horvat and Oliver Ekman-Larsson each had two assists, and Martin made 34 saves for the Canucks (12-12-3), who have won six of eight.
"Obviously, the way we started [the season], that's in the past," Pettersson said. "We've just got to learn from it ... and, hopefully, we can just keep it going and keep on building."
Logan Couture had a goal and an assist, and Hertl had two assists for the Sharks (8-16-5), who have lost seven of eight (1-5-2).
"I liked a lot about our game," San Jose coach David Quinn said. "I know we lost, but I liked our resiliency. I like that we got better and better as the game went on. We were down 3-1 and we battled back."