Keller and Conor Garland each had a goal and an assist, Jakob Chychrun had two assists, and Darcy Kuemper made 32 saves for Arizona, which twice trailed by two goals.
"I think we actually played pretty well and guys played hard," Kessel said. "I thought it was OK for us."
John Leonard, playing on a line with Hertl and Kane, had two assists in his NHL debut. Ryan Donato and Logan Couture scored in the shootout, when Jones made the save on two of Arizona's three shooters.
"It's just part of the game, you know that's something you deal with as a goaltender," Jones said of allowing Kessel's late tying goal. "You learn how to move past goals, and there's no time to really feel sorry for yourself because we're in a tight game with a chance to win."
It was the Sharks' first game in 309 days. They were not among the 24 teams in the NHL Return to Play Plan after last season was paused March 12 due to concerns surrounding the coronavirus.
"It's our first game in a long time. Some guys were playing their first NHL games and they were terrific. To play a game in this situation, I thought we did did very well," Couture said.