Fox went forehand to backhand and roofed a shot blocker side past Jonathan Quick to give the Rangers the win in the sixth round of the shootout.
"I think he should be one of our top shooters," New York coach Gerard Gallant said. "I think after tonight's goal he's going to get a lot of confidence from it. He's got skill, he's got talent and he made an incredible move. I talked to him earlier in the year about going [in the shootout] and he said, 'No, I don't feel comfortable going.' I think that's going to change now."
Fox said it was his first shootout attempt since he was a teenager playing in the USA Hockey National Team Development Program.
"I was debating between two [moves], stuck with that one and glad I chose it," Fox said. "The one I did I'm comfortable doing that. I didn't want to overcomplicate anything, so I just made a little fake and I'm glad it worked out."
Chris Kreider became the first NHL player to score 30 goals this season, Artemi Panarin had two assists, and Igor Shesterkin made 34 saves for the Rangers (28-11-4), who have won five in a row at home and five of their past six games overall.
Alex Iafallo and Blake Lizotte scored, and Quick made 28 saves for the Kings (21-16-6), who were playing the second game of a season-long six-game road trip. They defeated the New Jersey Devils 3-2 at Prudential Center on Sunday.
"It's bittersweet," Los Angeles captain Anze Kopitar said. "Yeah, three out of four [points] is OK. I guess if it was the other way around, we got one yesterday and two today I would have been a little bit more happy sitting with you guys (members of the media) here. But in the grand scheme of things, if we're going to go three out of four the rest of the trip, it's going to be a pretty good trip."