Williams and Anze Kopitar each had two goals and an assist, and Jonathan Quick made 25 saves as the Kings became just the ninth team in NHL history to force a seventh game after losing the first three.
Stalock stopped 26 shots in his first NHL playoff start, and James Sheppard scored for the spiraling Sharks.
Just three teams in NHL history have rallied from an 0-3 deficit to win a series.
Los Angeles could join them in Game 7 in San Jose on Wednesday.
I'm here to develop a winning culture and a winning team. There is a bright future ahead of this team. A lot of young, great hockey players here and on the way … a fan base that's passionate about the game of hockey.
— Dan Bylsma after being introduced as new coach of the Buffalo Sabres