Burakovsky scored on a power play at 2:08 of the extra session to give Seattle the first six-game winning streak in franchise history.
"End of the day, it was one of those games where last shot wins, and that's exactly what happened," Kraken coach Dave Hakstol said. "It's better to be on the winning side and the good side of it than the other, but it's a crazy hockey game.''
It was the sixth game in the expansion era (since 1967-68) to feature at least 17 goals and be decided by one goal and just the second since 1984-85. The other was when the Winnipeg Jets won 9-8 at the Philadelphia Flyers on Oct. 27, 2011.
Matty Beniers and Jared McCann each scored twice as the Kraken, playing their second NHL season, set a team record for goals. Alex Wennberg, Daniel Sprong and Oliver Bjorkstrand also scored, and Martin Jones made 27 saves.
Gabe Vilardi had two goals for the Kings, but Cal Petersen allowed four on 16 shots after replacing Jonathan Quick early in the second period. Anze Kopitar, Viktor Arvidsson, Carl Grundstrom, Adrian Kempe, Sean Durzi and Mikey Anderson also scored for Los Angeles.
Durzi cut it to 8-7 with 8:16 remaining in the third period, and Anderson tied it with 6:21 to go.
"It's a very, very confusing night for all of us,'' Kings coach Todd McLellan said. "I'm really disappointed that we gave up nine goals, absolutely. But on the other side of the coin, I'm glad that we fought back, got a point, and scored eight. So it's mass confusion right now for all of us, and we better fricking figure it out quickly.''
It looked as if the Kings were in position to score in overtime when Trevor Moore played the puck in the offensive zone before Kempe could get to the bench, resulting in a penalty for too many men on the ice. Burakovsky fired a wrist shot from the right circle through traffic on the ensuing power play.
"I got the puck in a good area, and I'll take that shot from there all day,'' Burakovsky said.
Despite securing the win with his second multigoal game in the past three outings, Burakovsky wasn't happy.
"It was kind of a messed-up night,'' he said. "We were trading chances, a lot of sloppy plays. I think we played a really bad game, and so did they. We were lucky that they didn't play good at all.''
Seattle topped the eight goals it scored against San Jose last Wednesday. McCann likened the game to playing pond hockey growing up in Canada before then pointing out the similarities to junior hockey.
"It's going to be one of those nights,'' he said. "Going to be a lot of goals, going to be a lot of offense, so good to get the two points.''
The first period was practically a game unto itself, producing a 3-3 tie after 20 minutes with Kopitar scoring 16 seconds in. It was his 11th career goal in the first minute of a game, which puts Kopitar sixth in NHL history, and it set the stage for two power-play goals per team on their four combined opportunities. Kraken defenseman Justin Schultz picked up the secondary assist on all three of his club's goals to set a franchise record for most assists in a period.
ESPN Stats & Information and The Associated Press contributed to this report.