Fleury gave Anaheim a 1-0 lead at 5:37 of the first period with his first goal since being acquired in a trade from the Carolina Hurricanes on April 12. Rookie center Trevor Zegras won a face-off to Fleury at the left point for a shot through traffic.
It was Fleury's second goal in 43 games this season.
"All the guys, all the coaches are really welcoming and really encouraging me to play how I know I can play," he said. "[Scoring] is something I feel I can do, and that's something I think I can improve in my game."
The Ducks outshot the Kings 20-6 in the second period.
"Your goaltending is going to have to stand tall on nights where you're just not generating anything, but on a night like tonight, we should've had run support," Eakins said. "You're rolling that third period, and it would be way better to be up by two or three than hanging on by one."
Quick made 10 saves shorthanded, helping Los Angeles go 4-for-4 on the penalty kill.
"I don't think you have to be a rocket scientist to see that he was the reason we were still in the game," McLellan said. "He was outstanding. Some of the saves he made, and secondary chances he somehow got a pad on or a blocker. He was their savior for two periods."
NOTES: Kopitar has 345 goals and 652 assists in 1,121 NHL games, all with the Kings. ... Los Angeles forward Quinton Byfield, the No. 2 pick in the 2020 NHL Draft, was back in the lineup after making his NHL debut against Anaheim on Wednesday. He had one shot on goal and two hits in 17:18. ... Kings forward Blake Lizotte sat out with an upper-body injury sustained against the Ducks on Wednesday. ... Zegras has three assists in three games since he was recalled from San Diego of the American Hockey League. ... Ducks forward Max Comtois had an NHL career-high seven shots on goal. ... Anaheim forward Ryan Getzlaf did not play because of a back injury.