What we learned from the Kings’ 2-1 overtime win over San Jose

Here’s what we learned from the Kings’ 2-1 overtime victory over the Sharks on Tuesday at San Jose:

Peter Budaj has gone from the waiver wire to season-saver.

On Oct. 3 the Kings, secure with Jonathan Quick as their starting goalie and Jeff Zatkoff as their backup, waived Budaj so they could send him to the American Hockey League. Exactly three months later, with Quick on injured reserve and Zatkoff having lost the confidence of Coach Darryl Sutter, Budaj made his 33rd appearance in 38 games and came up with 26 saves. He was fortunate at times — Melker Karlsson, shooting from one knee, was unable to get good wood on a point-blank shot that Budaj stopped at 8:49 of the third period — but he also was good when the Kings needed him to be as they swept a home-and-home sequence from their division rivals.

“It’s great. We were in a little slump there before,” Budaj said. “We lost in Vancouver and we lost in Edmonton and we just knew that we had to come out and do a little bit better and do what it takes to try to win. It’s a very tough opponent that we played and it’s a good two points — good four points — for us.”

Teammate Drew Doughty praised Budaj for compensating for the team’s defensive mistakes. “There were a couple of looks that we never should have given up and he had to make big saves,” Doughty said. “We can’t have any lapses at any time. When you’re playing in the National Hockey League, you make one little mistake it’s usually going to bite you in the butt and be in the net. So it was great that Boods stepped up for us and made those saves. But we’ve got to clean that up.”

Budaj said the heavy workload hasn’t tired him out, which is good news for the Kings because Quick isn’t expected back anytime soon.

“Thank God I’m feeling good,” Budaj said. “I’m very happy that I can play this many games and help the team to win games. I’m just very excited, and I will keep working hard and just enjoy this moment.”

Special teams did special work.

The Kings’ penalty killers were three for three, the first time in four games they hadn’t allowed an opponent to score a power-play goal. And the power play was one for three, connecting on Jeff Carter’s redirection of a Jake Muzzin pass in the third period. Their penalty killing has an 83.6% success rate, 10th in the NHL. Their power play, however, ranks 22nd with a success rate of 16.1%. That could use some work.

The Kings can rise to the occasion.

The idea they would sweep a home-and-home series from the Sharks seemed unlikely last week, when they ended a stretch of nine straight road games with losses at Dallas, Vancouver and Edmonton and had to put right wing Tyler Toffoli on injured reserve after he suffered an upper-body injury.

But faced with the prospect of dropping back in the West playoff picture, the Kings pulled out one-goal wins at home and on the road against the Pacific-leading Sharks. In some games, they look like Stanley Cup contenders. In their next game they might look like they deserve relegation to the American Hockey League. They’ve gone back and forth most of this season so far and it’s baffling, though entertaining.

Jeff Carter carried them against San Jose. He scored the winner in a 3-2 decision at Staples Center on New Year’s Eve and then scored the tying goal in the third period on Tuesday before setting up Tanner Pearson’s winner 58 seconds into overtime. He also was 9-8 in the faceoff circle.

But in addition to Carter’s exploits and their solid penalty killing the Kings did a lot of things right on Tuesday, including holding San Jose defenseman Brett Burns — the NHL’s shots leader — to a single shot on goal. Doughty excelled in his 28 minutes and 48 seconds of ice time and is tied for second in the NHL in average ice time, at 27:11. (Winnipeg’s Dustin Byfuglien leads at 27:18.)

It’s tough to predict whether the good Kings or the bad Kings will show up next, but they’ll need to show their good side more consistently as they settle in for seven straight home games.

helene.elliott@latimes.com

Follow Helene Elliott on Twitter @helenenothelen