Dwight King and Alec Martinez also tallied and Dustin Brown and Matt Greene contributed two assists apiece as Los Angeles won for only the second time in seven games.
Bernier had 10 saves in each of the first two periods en route to his fifth career shutout and second against Minnesota.
Williams triggered a three-goal first period when a shot by Slava Voynov deflected off his body and past Niklas Backstrom only 44 seconds into the game.
King doubled the lead by redirecting a centering pass from Kopitar nine minutes later, and Kopitar ended a nine-game goalless drought by tapping a shot into an open net with 1:22 left in the period.
Martinez supplied the only goal of the second period to give Los Angeles a commanding 4-0 lead.
GAME NOTEBOOK: The Kings, who entered with a league-low 130 goals, had been limited to one goal or fewer in eight of the previous 12 games. The four-goal outburst allowed them to surpass the Wild, who came in with 133 goals. … Bernier won for the first time since Jan. 9. It was only his second start since that victory. He also blanked the Wild in Minnesota on Feb. 1, 2011.
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