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.
The groove of being behind a bench is going to be interesting at first, but thank God we have a few exhibition games to get rid of those cobwebs. Overall the excitement of it all and the freshness and coming back refreshed, all those things are going to be assets. If [the players] come ready to give their best effort in practice and games, good things are going to happen. I'm always looking for results. It's not always on the scoreboard. It's winning and building something.
— Bryan Trottier on making his return to coaching as an assistant with the Sabres