The Kings dominated early despite a week off and outshot Phoenix 48-27.
Anze Kopitar scored 4 minutes in, Dustin Brown had a goal and an assist, and King had his first career two-goal game to give Los Angeles its sixth consecutive road victory of the playoffs.
Derek Morris beat Jonathan Quick from the center line, and Mikkel Boedker also scored for the Coyotes, who trail in a series for the first time this year.
Game 2 is Tuesday night in the desert.
This was the Western Conference finals few expected.
The Coyotes weren’t picked to even make the playoffs, entering the season with an unproven No. 1 goalie and no owner for the third consecutive year.
Grinding and relying on Mike Smith’s breakout season in goal, the Coyotes won their first division title as an NHL franchise and beat the Chicago Blackhawks and the Nashville Predators, two supposedly better teams, in the first two rounds to make their inaugural appearance in the conference finals.
The Kings had expectations, but didn’t quite live up to them.
Los Angeles was among the NHL’s worst-scoring teams all season, costing coach Terry Murray his job in December, and squeaked into the playoffs as the last team in the West behind Quick’s stellar season.
The Kings, too, were underdogs in the playoffs and didn’t seem to mind, racing through the Vancouver Canucks and the St. Louis Blues to reach the conference finals for the first time since 1993 and become the first No. 8 seed to knock off the top two teams in the same playoffs.