Troy Terry and Shattenkirk scored unassisted goals in the second period and John Gibson made 30 saves before stopping two of Los Angeles' three tries in the shootout. The Ducks got their 10th win in 14 games despite wasting a 4-1 lead with 13 minutes left in regulation after Cam Fowler and Isac Lundestrm scored 29 seconds apart early in the third.
"It's hard to walk out of road games with two points, but we're not nave to the fact that there's some things we've got to clean up to try and play better hockey with the lead, and continue to put pressure on teams and not sit back," said Fowler, who put a backhand off the post in overtime. "Credit to them for throwing everything they had at us in the third."
Jonathan Quick stopped 32 shots and Dustin Brown scored the tying goal with 4:26 left in regulation for the Kings, who have lost six of seven after a seven-game winning streak.
Alex Iafallo scored two goals for Los Angeles, and Adrian Kempe started the third-period comeback with his eighth of the season.
"It doesn't happen often, but momentum is a strange thing," Kings coach Todd McLellan said. "You can gain it or lose it quickly, and I've been on the other side of something like that where it just gets away on you and it's like a big snowball just rolling down the hill. Fortunately for us, it was in our favor, but we couldn't finish the night."
Lias Andersson scored in the first round of the shootout for Los Angeles, but Zegras tied it for Anaheim in the second. Gibson stopped Anze Kopitar before Shattenkirk beat Quick for the winner.
A lively Staples Center crowd was rewarded with a barn-burner of a game when these revitalized Southern California rivals met for the 153rd time. Both clubs are showing significant progress this season after bottoming out with concurrent three-year playoff droughts.
Anaheim has been competitive near the top of the Pacific Division this season despite the departure of general manager Bob Murray, while Los Angeles has taken clear strides with new talent around its Stanley Cup-winning cornerstones.
"Hey, that was an exciting hockey game," Anaheim coach Dallas Eakins said. "Two teams kind of trying to do the same thing at the same time, and I thought both teams worked extremely hard tonight. That was a real hardworking hockey game by the Kings and by us."
Drew Doughty had an assist while playing almost 28 minutes in a quick return from seriously bruising his knee on Oct. 22. The Kings initially projected Doughty would be out for two months, but the 2016 Norris Trophy winner (and famously quick healer) missed five weeks and 16 games with the most significant injury of his 14-year career.
"I was a little worried before my first couple shifts, not gonna lie," Doughty said. "I haven't been nervous before a hockey game probably since my first NHL game, and tonight I kind of felt it. Hadn't had to play through an injury like this before, so had a little butterflies, but I felt good after my first few shifts."
Anaheim captain Ryan Getzlaf left the ice in the first period with an apparent left ankle injury. The 36-year-old center is in the midst of an impressive bounce-back season with 20 points - including the 1,000th of his career - in 23 games. He will be re-evaluated Wednesday.
"It's never good when a guy doesn't come back in a game," Eakins said. "Especially him, because he's as tough as they come."
After Terry opened the scoring with the 14th goal of his breakout season, Anaheim went up 4-1 early in the third. Shortly after Fowler got a power-play goal that deflected off Trevor Moore, Lundestrom got credit for a goal when Los Angeles' Matt Roy accidentally backhanded a clearance attempt off teammate Alex Edler and into Quick's net.
"I mean, like two of those goals were pure luck," Doughty said. "Their third and fourth goal, they're both deflections. One we put off our own foot and in. So we stayed positive."
Kempe scored on a deflection to make it 4-2, and the Kings initially appeared to get another 21 seconds later before the officials waved off Carl Grundstrom's goal for a high stick. Los Angeles kept up its pressure, and Brown followed Iafallo's second goal by tying it with his third of the season.
"I guess we technically scored five, but one got disallowed," Doughty said. "That could have been the game right there."
Ducks: Host Vegas on Wednesday night.
Kings: Host Calgary on Thursday night.
Shattenkirk wins it for the Ducks in a shootout
Kevin Shattenkirk scores the deciding shootout goal to lift the Ducks past the Kings.