Dustin Brown and Dwight King scored third-period goals to clinch the Game One victory, with the second game of the best-of-seven series set for Tuesday, again in Phoenix.
The win continued a fairytale post-season run for the West's eighth seeds, who have won nine of 10 play-off games to eliminate top seeds Vancouver in the opening round then swept the second-seeded St Louis Blues in the second.
"It was really important we had a good start," Kings coach Darryl Sutter told reporters. "It wasn't so much scoring or any of that stuff.
"Just make sure we match their start."
Unbeaten on the road during the playoffs, the Kings spoiled the NHL-owned Coyotes' first appearance in a conference finals.
Phoenix netminder Mike Smith kept the home team in the game in the first period where they were outshot 17-4 by the Kings but went into the intermission tied at 1-1.
However, he could not withstand the relentless Los Angeles barrage and surrendered two more goals. The Kings finished with a total of 47 shots.
"We knew coming in, you watch Phoenix play, they got a really good goaltender," said Brown. "Part of our thing is getting as many shots on him as we can, make him work, make it hard for him."
Anze Kopitar gave the Kings an early lead with his fourth of the postseason before Derek Morris answered for the Coyotes.
King restored Los Angeles' lead midway through the second only to have Mikkel Boedker answer for Phoenix to send the teams into the second intermission deadlocked at 2-2.
Los Angeles captain Brown notched the game winner just over two minutes into the third and King collected his second of the night into an empty net with 48 seconds left.
"It's just like pretty much any series," said Brown. "You get that first game out of the way, start looking at film, tendencies.
"I'm not sure what they're going to try to do to stop us. It's one of those things where they're going to make adjustments.
"It's up to us as players to make adjustments in games to be effective and have an impact on Game Two."