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 sure [my father] was going crazy up there. I hope all of my friends were celebrating in the crowd. Coming back here and all of my friends and family are here, getting a goal in front of them is special.
— Blues forward Ryan Reaves on scoring the game-winning goal Sunday against the Jets in his hometown of Winnipeg