Ersberg earned his second career shutout in his 10th NHL start, all in the past month. Ersberg also shut out the Ottawa Senators 2-0 on March 6.
The Coyotes had won the first six meetings between the Pacific Division rivals this season, but the Kings took the final two games, both with Ersberg in goal. The 26-year-old rookie keyed a 6-5 shootout win in Phoenix on March 20.
"You have to give him credit, he played well last game, he played well tonight," Phoenix captain Shane Doan said. "He's a good goalie. You have to get a little bit more traffic in front of the net, but you have to give him credit."
"When you get the first goal it gives you wings and you are more comfortable making plays you normally wouldn't," Kopitar said. "We have really quick defensemen and when they join the attack it really helps us."
Scoring goals has not been a problem for the Kings this season, dependable goaltending has been. Ersberg is making a strong case he is ready to change that.
"Erik has been playing really good, we are really comfortable with him right now," Kopitar said. "The last couple of games, he has really bailed us out."
Ersberg has won three of his past four starts, with the only defeat being a 2-1 shootout loss at Anaheim on Wednesday in which he made 39 saves. He has stopped 102 of his past 105 shots and held the opposition to two goals or fewer in seven of 10 starts since being recalled from Manchester of the American Hockey League in late February.
"It feels great, I couldn't ask for much more," said Ersberg, whose parents had traveled from Sweden to attend Thursday's game. "I think we played a great game defensively. I saw almost every shot and (the defense) was there on the rebounds too, so they helped me out a lot."
Ersberg made a right pad save against Peter Mueller on a breakaway to keep the game scoreless through one period.
Dallman gathered a cross-zone pass from Kopitar and swiped the puck past Phoenix goalie Mikael Tellqvist to give the Kings a 1-0 lead 6:52 into the second.
Just 2:22 later, Kopitar backhanded in a rebound of a Brian Willsie shot for his 30th goal of the season, making it 2-0. The Kings completed the scoring run with Brown getting his 32nd goal of the season during a power play 13:03 into the period.
The Coyotes have just one win in their past seven games.
"We came out and played a pretty good first period, and then laid an egg in the second and third periods," Coyotes defenseman Ed Jovanoski said. "We hung our goalie out to dry, and that's unacceptable."
Kings goalie Dan Cloutier was forced out of the lineup earlier Thursday with a right groin strain. The team recalled Daniel Taylor from Manchester on an emergency basis to back up Ersberg.