The Kings rallied to win the best-of-seven series after losing the first three games, winning four in a row.
Anze Kopitar scored the tiebreaking goal late in the second period and Jonathan Quick made 39 saves to cap the Kings' historic comeback.
Drew Doughty, Tyler Toffoli, Dustin Brown and Tanner Pearson also scored for the Kings, who became the fourth NHL team to win a best-of-seven series after losing the first three games.
Matt Irwin scored the lone goal and Antti Niemi made 25 saves for the Sharks.
The Kings will meet the Anaheim Ducks in the second round of the playoffs, the first time the two teams have met in the playoffs.
The Associated Press contributed to this report.