Williams ended a choppy overtime period by beating Rangers goalie Henrik Lundqvist high with his eighth goal of the playoffs after a turnover by New York defenceman Dan Girardi.
Williams scored his seventh career Game 7 goal, tying a record, to put the Kings past Chicago in the previous round.
New York caught Los Angeles off guard with their speed up front in the first period of the Cup opener, opening up a 2-0 lead at the 15:03 mark of the first on goals by Benoit Pouliot and Carl Hagelin.
The Kings got one back courtesy of Kyle Clifford late in the first, and Drew Doughty tied the contest for Los Angeles just over six minutes into the second.
Los Angeles dominated play in the third with an 18-3 advantage in shots. In the final minute, Hagelin and Jeff Carter of the Kings each had near misses.
Lundqvist finished with 40 saves, while Jonathan Quick stopped 25 shots for the Kings, who are looking for a second championship in three years.
Game 2 takes place at Staples Center in Los Angeles on Saturday, with Hockey Night in Canada coverage beginning on CBC-TV and CBCSports.ca at 6:30 p.m. ET with Hockey Tonight.
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