Kopitar scored his first goal of the season when he tipped in Kempe's shot to make it 4-1 at 8:22.
Lias Andersson scored on a breakaway to make it 5-1 at 17:18.
"I would say without a doubt, all three zones were solid," Kings coach Todd McLellan said. "We were skating. There was a tenacity to us. We were responsible. I thought this was the first time we took the game to a team rather than receiving it."
St. Louis scored two goals in 48 seconds to cut the Los Angeles lead to two. Schenn's second of the game made it 5-2 at 6:01 of the third period, and Ryan O'Reilly scored from the point to pull the Blues within 5-3 at 6:49.
"We needed to come out and respond and get to our game, but I think we're overthinking it and we get caught on our heels," said O'Reilly, the Blues captain. "You can just kind of see it. It feels like every bounce goes their way and we're creating those bounces going against us because we've overthinking and we're not going and just trusting ourselves. That's what happens and you can see in the third period when we do kind of trust each other and we just go, go, go, bounces start to come our way and there's chances and we have a push. It's disappointing, it's the thing we've been struggling with the whole season so far. It's obviously a concern. We have to find a way to do that."