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.
It's really exciting. I'm pretty sure that when I play my first game I'm going to be emotional. To be back on the ice playing a game, being in game situations, with all the routines and rituals I do before games and during the game, I feel like I'm going to be emotional. I'm going to be really happy.
— Montreal Canadiens forward Tim Bozon on playing for the first time since his life-threaning illness