Kane's second goal late in the third period had seemingly put the Blackhawks in the driver's seat 3-2 with less than 4 minutes left in regulation.
But the defending NHL champion Kings forced overtime when Mike Richards redirected an Anze Kopitar shot into the net with 9.4 seconds left.
After a scoreless first overtime, Chicago got the puck on a Los Angeles turnover with Kane and Jonathan Toews coming down the ice on a 2-on-1 break. Toews fed a precise pass to Kane, who blasted it over the glove of Jonathan Quick for the game-winner.
Duncan Keith also scored while Toews and Bryan Bickell each added two assists for the top-seeded Blackhawks, who won the Western Conference finals 4-1 and now go on to face the Boston Bruins for the Stanley Cup title.
Corey Crawford turned aside 33-of-36 shots for the winners.
Kopitar scored and Quick finished with 31 saves for the fifth-seeded Kings, whose goal of back-to-back NHL championships came to an end.