Los Angeles snapped a 14-game streak in which it did not scored more than two goals in regulation, a slump that dated to before Thanksgiving.
The Kings moved to 2-0-1 under new coach Darryl Sutter as the NHL's lowest-scoring team got unlikely contributions from Mitchell, Rob Scuderi and Brad Richardson.
"That's what it's going to take," Brown said. "We're going to have to have scoring from the back end. Tonight we had a lot of shots. We had some grade-A scoring chances going to the net. We hit three or four posts. We've been getting more shots on net, and tonight they started to go in. I think we can continue with that mentality. You're going to see us score more goals."
Mitchell wristed in a shot from the right side fewer than two minutes into the third, and Brown had his shot trickle between the pads of Jason LaBarbera at 7:06.
L.A. had 35 shots on goal and got a great start in which it scored two goals in the first period for only the second time this season.
"Nice to see a 'W'," Sutter said. "That's more important than (scoring four goals)."
Sutter has the Kings playing the same system, and the players say their new coach has changed them in terms of approach.
"The biggest thing here – it's just an attitude," Scuderi said. "We're trying to turn the page. It's unfortunate it costs someone their job. But we have to try to move on. I think the play will clean itself up. Right now we're trying to stay positive and so far it's working for us."
The one drawback for the Kings was that they might have lost winger Simon Gagne, who did not play in the third period because of a possible head injury.
Sutter said Gagne got "banged up in front" and will be evaluated in the morning.
The Coyotes had plenty injury issues entering Monday's action. Goalie Mike Smith missed a third straight game with a groin injury, and the Coyotes literally had to patch a big hole in their lineup because centers Martin Hanzal and Boyd Gordon sat out with upper- and lower-body injuries, respectively.
But Phoenix mostly held its own with two goals by fourth line wing Raffi Torres and a contribution from newly-recalled Patrick O'Sullivan. Torres took a long stretch pass from O'Sullivan and scored a breakaway goal to tie it at 2 midway through the second, a sloppy period by L.A. in which it was charged with nine giveaways.
"You could call it cherry picking," Torres said. "I didn't expect much, but Sully put it right on my tape in mid-stride."
"I thought we hung in there," Phoenix coach Dave Tippett said. "We wanted to make sure we competed hard and give ourselves a chance of winning. We were hanging around the whole game right to the end."
But the Coyotes, who took a Western Conference-best 11-6-1 road record into Monday, opened a post-Christmas road stretch with a loss.
"Everyone has injuries, everyone has people that are missing," captain Shane Doan said. "You have to find ways to battle through it and make sure that other guys step in. I thought Raffi Torres was great and O'Sullivan. That line was really good for us. We had guys step up, we just to find ways to score on the power play. Special teams this year are huge for us and we got to be better at it."
The Kings took a 2-1 lead on simple, hard work by their forwards.
Richardson brought the puck across the blue line for an extended period in the Phoenix zone and then went in front of the net to tip Slava Voynov’s waist-high slap shot from the right side at 13:48.
Scuderi scored with a slap shot from the right point at 4:09. Kopitar started the whole sequence when he won a puck battle to keep it in the Coyotes' zone.
"The first minute's the most important minute of every hour," Sutter said of the solid start.
Phoenix responded to Scuderi's goal with a terrific play by Torres, who grabbed a loose puck in the Phoenix end, skated down and flung the puck over Quick's shoulder with Jack Johnson on his back. Hanzal got high sticked in the face early in the third period last Friday against St. Louis.
The Coyotes on Monday recalled O’ Sullivan and goalie Curtis McElhinney to back up LaBarbera. McElhinney relieved LaBarbera after Brown's goal.