Clayton Keller, Lawson Crouse and Anton Stralman scored for Coyotes (12-33-4), who have lost nine of their past 11 games. Karel Vejmelka made 34 saves for Arizona, which had been 6-0-0 when leading after two periods.
"They scored, made it 3-2, and that put us back on our heels," Coyotes coach Andre Tourigny said. "And I didn't like the way we played the third. I thought we froze, kind of, and instead of digging deep, we forced things."
Keller gave Arizona its 3-1 lead at 10:40 of the second period with a wrist shot off a rush before the Kings began their second comeback in as many days. They trailed 2-0 against Vegas on Friday.
"Going up there, we know they're going to have a push," Coyotes forward Barrett Hayton said. "I think just we've got to manage that better."
Kings coach Todd McLellan sensed some frustration when the Kings couldn't convert on several quality scoring chances as Arizona built its lead.
"You tighten up a little bit more because it's not going in and, before you know it, you've made some some mistakes on judgment situations, and it's in your net," McLellan said. "The good thing is that we could reel it back in and stick with the plan … and leave here with points."
Doughty cut it to 3-2 at 18:33 of the second with a power-play goal, a one-timer off Kempe's pass from the right circle.
"We were jawing back and forth in the locker room about the nets he missed," McLellan said. "He could have had four [goals]. He probably could have had his first hat trick, but we'll take two from him. He's playing tremendous."
Trevor Moore tied it 3-3 at 6:24 of the third period off the rebound of Viktor Arvidsson's shot. Doughty scored 48 seconds later to make it 4-3.