Curry hit a play-off career-high 50 points as his side won 120-100.
Earlier, the Miami Heat beat the New York Knicks 108-101 to go 1-0 up in the Eastern Conference semis.
The Warriors were trounced 118-99 in game six on Friday but Curry, chasing a fifth NBA title in nine years, starred in the decider – finishing with the most points ever in a game seven – as his side progressed.
“What an incredible all-time performance,” said Curry’s team-mate Klay Thompson. “This is a game seven I’ll forever remember as the Steph Curry game.”
“We all take him for granted because he’s brilliant night after night and we’ve been watching this for 10 years,” added Warriors head coach Steve Kerr.
“You just have to remind yourself, big picture, this is one of the best players in the history of the game.”
The Kings, who were making their first post-season appearance in 17 years, and had the benefit of a home crowd, led 58-56 at half-time with Domantas Sabonis scoring 16 points.
But the third quarter belonged to the Warriors, and once Curry scored five points in a row to put them 69-62 up with seven minutes and 50 seconds remaining they seized control and never relinquished it.
Curry was devastating with seven three-pointers in his tally but Kevon Looney (21 rebounds and 11 points), Andrew Wiggins (17 points) and Klay Thompson (16 points) also played their parts in the win.
The Warriors faced a James-led Cleveland Cavaliers in four straight NBA finals between 2015 and 2018, winning three of them. Four-time champion James won his last title with the Lakers in 2020.
Speaking about facing NBA record scorer James again, Curry added: “It’s special from the first series we played him in Cleveland. We’re blessed to be playing at this level still and excited about a new chapter.”