Vilardi's second goal gave the Kings a 3-2 lead 4:42 into the third period. Lias Andersson carried the puck beyond the goal line and made a backhand pass to Vilardi in the low slot.
Kopitar won a face-off to set up Walker, who scored into an empty net at 19:23 for the 4-2 final.
"It's incredible for him. We're all along for the ride," Los Angeles coach Todd McLellan said. "He makes everybody better, including the coaching staff, and his passion and energy for the game are contagious. … I can't wait to see him get the next 500."
The 33-year-old center is the fourth Kings player with 1,000 points, joining Marcel Dionne (1,307), Luc Robitaille (1,154) and Dave Taylor (1,069). He has 346 goals and 654 assists in 1,124 games in 15 seasons with the Kings.
"It's hard to put into words," Kopitar said. "I'm kind of speechless right now. I'm obviously very excited, very happy and, you know, I was trying to do it a little bit earlier in the game, but I guess it worked out the way it worked out."
Kopitar is one of three players with 1,000 points and two Selke Trophy wins (Sergei Fedorov, Rod Brind'Amour) and the 31st to accomplish it with one team.
"I've said all along that makes it more special," Kopitar said. "It's great to be part of this club, and hopefully there's a few more to come."
Arizona coach Rick Tocchet had mentioned how dangerous Kopitar is on face-offs.
"He's one of my favorite players," Tocchet said. "He plays the game the right way. If he cheated a little bit more [defensively] he could probably get 15 to 20 points more a year. I know he could. But he plays a 200-foot game and takes every important draw. He's classy and I love the way he plays the game."
Los Angeles swept the two-game set to finish 5-3-0 against Arizona this season. The Coyotes were 3-4-1.
Ekman-Larsson's wrist shot from the point while Petersen was screened gave the Coyotes a 1-0 lead at 1:19 of the first.