Kane's goal started a three-goal barrage for San Jose in the first period with Rudolfs Balcers and Kevin Labanc following up to help San Jose sweep the two-game series against Arizona in their first two games in front of their home fans.
The Sharks were allowed 520 fans on Monday and 1,037 for this game.
"It's getting there," captain Logan Couture said when asked if it's feeling like a normal home game again. "The last two games have been very nice. Our fans are awesome. We love the support, We wish there could be more in the building. But both games they've done a great job. They are noticeable."
The Sharks entered the series on an eight-game losing streak that all but knocked them out of playoff contention.
With the back-to-back losses the Coyotes fell three points behind St. Louis for the final playoff spot in the West Division with the Blues holding three games in hand. Arizona has lost nine of 11 games overall.
"I think the last little while we've been making too many mistakes," captain Oliver Ekman-Larsson said. "That comes down on me. I don't know what I'm doing out there. We have to clean that up."
Tomas Hertl sealed the game with an empty-net goal for San Jose.
Phil Kessel and Johan Larsson scored but the Coyotes couldn't get anything else past the 23-year-old rookie Korenar. He got his first two career wins this week against Arizona.
"You could tell he looked a little more comfortable in there tonight," coach Bob Boughner said. "I know he wasn't happy with a couple of the goals the other night. It's good he came out and found his game."
Kane played a big role in both wins with a goal and three assists on Monday night and then another goal in this game to give him 20 on the season. Kane became the fourth player to reach at least 20 goals in each of the past six seasons, joining Alex Ovechkin, Brad Marchand and Aleksander Barkov.
Darcy Kuemper made 27 saves for Arizona.
The Sharks jumped on top of Arizona early for a second straight game. After scoring 17 seconds into the game on Monday, the Sharks took 1:12 to get on the board in this game when Kane finished off a nifty passing sequence started by Tomas Hertl and Alexander Barabanov.
San Jose added to that lead when Balcers took a turnover at the blue line and skated in to beat Kuemper with a backhand to make it 2-0.
Labanc added the third goal when he knocked in a loose puck after a scramble in front of the net.
"The start is killing us," Coyotes coach Rich Tocchet said. "Three gifts again. Wrong reads. I don't understand the reads. A couple of guys are fragile on these reads at the beginning of the game and it's in our net."
Kessel got the Coyotes on the board late in the first period when Nick Schmaltz's shot deflected off him and in for his 18th goal. Arizona had an extra skater on the ice for 59 seconds because of a delayed penalty before the goal. It was Kessel's eighth goal of the season against San Jose, tying a record for the most vs. the Sharks held by Teemu Selanne (1995-96) and Luc Robitaille (1992-93).
Coyotes: Host Vegas on Friday night.
Sharks: Visit Colorado on Friday night.
Labanc increases Sharks' lead with a goal in front
Kevin Labanc records his 11th goal of the season as San Jose defeats Arizona 4-2.