Kakko put the Rangers up 3-2 at 18:44, scoring from the top of the crease with Shesterkin off for an extra attacker during a delayed penalty.
"Started playing our game, and that's what we can do," Kakko said.
Fiala gave Los Angeles a 1-0 lead 35 seconds into the first period.
Vilardi scored on the power play to make it 2-0 at 6:01, tipping in Fiala's pass at the far post. It was his 11th goal in 22 games, breaking his previous NHL best of 10 goals in 54 games in 2020-21.
"Going in with the lead like that, 2-0 in the first, we've got to find ways to just keep going, you know," Fiala said. "It's no reason to step off the gas."
NOTES: It was Kreider's 36th game-winning goal, passing Steve Vickers and tying Adam Graves for seventh in Rangers history. Brian Leetch (37) is sixth. … Fox extended his point streak to nine games (four goals, 10 assists). It is the second-longest point streak of Fox's NHL career; he had at least one point in 12 straight games from March 17-April 8, 2021. … New York was 0-4-2 when trailing after the first period. … Kings center Anze Kopitar had two assists for his 266th NHL multipoint game. … Walker's goal was his first since May 13, 2021.