Gabriel Vilardi scored the deciding goal in the fourth round of the shootout and had an assist on Carter's goal.
"I didn't feel any pressure at all," Vilardi said. "I had a chance to finish it for the team, and I was lucky enough to do it. I thought we had another nice game, and 'Quickie' made some really nasty saves again."
Brown gave the Kings a 1-0 lead 4:21 into the game after Anze Kopitar chipped him the puck out of a scramble along the left boards. It was his fourth goal in four games.
Carter made it 2-0 at 11:50 when Kuemper accidentally knocked the puck across the goal line. The goal initially was waved off for goaltender interference, but McLellan successfully challenged.
Garland scored from the high slot at 1:00 to make it 2-1, and Kessel scored his first goal in 12 games off Tyler Pitlick's rebound at 5:51, seven seconds after a Coyotes power play ended, to tie it 2-2.
"We finally found our work ethic," Arizona forward Christian Dvorak said. "First period, we didn't have enough of that. We just need to have a better start. That's the reason we didn't get two points."
Kessel scored four goals in Arizona's first four games, but he hadn't scored since Jan. 20 against the Vegas Golden Knights.
"That was a big goal for him, hopefully he can break loose, get some goals," Tocchet said.
Kopitar and Adrian Kempe also scored in the shootout for Los Angeles. Dvorak and Nick Schmaltz scored for Arizona.
"McLellan
got his first assist with Arizona on Kessel's goal.