Vancouver held a 2-1 lead and were just 44 seconds from securing the road win when Russian defenseman Slava Voynov scored to send the game into overtime.
Following a scoreless extra period, Jeff Carter netted the only goal of the shootout, after also scoring in regulation, to propel the Kings to an improbable win.
Los Angeles also stunned the Canucks (2-2-2) in the 2012 post-season where the eighth-ranked Kings eliminated the Western Conference top seeds 4-1 in the first round of the playoffs.
"I think it's only good for tonight," Kings goaltender Jonathan Quick said. Quick made 21 saves then fended off three Vancouver shooters to seal the victory.
"You get two days in between games now and we've got to use our time properly to be able to take the momentum to Thursday night (against Nashville)."
In the regular season rematch between the teams, Zack Kassian and Alex Burrows helped the Canucks forge a 2-0 lead early in the second period but the home team rallied to wipe out the deficit.
Carter halved the deficit at 15:21 of the second period and Los Angeles plugged away until they tied the game in the final minute.
The win was the second in a row for the Kings (2-2-1), who opened the lockout-reduced season with three consecutive losses.
Roberto Luongo made 26 saves for Vancouver but was less than happy with the outcome of the game.
"It's disappointing getting scored on in the last minute like that," Luongo said.
"(You) always want to find ways to be better, maybe make an extra save somewhere."