Alexis Lafreniere put New York ahead 1-0 at 14:00 of the first period, scoring from a sharp angle off a give-and-go with Kaapo Kakko.
Trocheck made it 2-0 at 2:02 of the second period when his centering pass intended for Chris Kreider redirected in off the stick of Kings defenseman Matt Roy.
Trocheck then beat Quick blocker side from the high slot in between four defenders to push it to 3-0 at 3:37.
"Whenever you get up, give yourself the momentum, you don't have to worry about digging yourself out of a hole like we've been doing a little bit lately," Trocheck said. "So to get out to a lead early was big for us."
Viktor Arvidsson cut it to 3-1 at 18:12 when he batted in the puck near the left post for a power-play goal.
Roy got the Kings to within 3-2 at 1:27 of the third period, finishing off a 2-on-1 with Phillip Danault with a wrist shot from the edge of the right circle.
Panarin quickly responded for the Rangers to make it 4-2 at 2:11. He put in his own rebound after Copley lost his balance making the initial save.
"It was just a couple tough bounces really," Los Angeles captain Anze Kopitar said. "I think there's not a whole lot of stuff that we didn't like in the first period or in the second. Obviously, sometimes bounces don't go your way. We realize you've got to play for your bounces, too, and tonight it didn't go for us."
Zibanejad made it 5-2 at 5:26 on a power play. He scored his 31st goal on a one-timer from the left circle off a pass from Adam Fox.