Trailing 2-1 with two minutes left, the Panthers scored twice in a 20-second span, and Pavel Bure capped the rally with an empty-net goal to deny the Kings a third straight road victory.
Kozlov scored from just outside the left crease on a feed from Ray Whitney to tie the game with 1:51 left.
Bure, who has 31 goals in 37 career games against the Kings, finished off the scoring with 34 seconds left as Florida won for the second time in as many games.
"That was a great win because it reinforces our system and what we're trying to do," Florida coach Terry Murray said.
"Whenever you have Pavel Bure in the lineup, that type of superstar, you believe you can do everything."
Luc Robitaille scored his third goal of the season and 520th of his career, but the Kings still missed their best start since 1988.
The Kings are in the middle of a seven-game road trip to start the season as they await the opening of the $375 million Staples Center.
"It was a very difficult game for us to lose, but it was our mistakes that cost us, and we're the ones who allowed it to happen," Los Angeles coach Andy Murray said.
"We knew it would be a 60-minute game, and it turned out to be a 58-minute game for us."
Ziggy Palffy scored his second goal of the season to put Los Angeles ahead 2-1 at 1:11 of the third period.
Palffy scored from the left side of the crease against Florida goalie Trevor Kidd, starting in place of Sean Burke, who is out with a concussion from Saturday's opening win against Washington.
The lead held up until the final two minutes.
Ray Whitney set up Kozlov's tying goal with a pass into the slot. Kozlov knocked it in to cap the 2-on-1 break.
"All I had to do was hit the puck," Kozlov said. "It feels good because it seems we haven't won a game like this in two-three years."
The Panthers scored again 20 seconds later when Scott Mellanby dug the puck out from behind the net and sent it to the blue line.
Hedican fired it toward the goal, and Parrish tipped it in for the game-winner.
"I didn't see it," Fiset said. "I thought it was going wide, but it looked like it hit somebody out in front."
Florida led 1-0 in the second period on Robert Svehla's goal at 5:53, but the Kings tied it on Robitaille's goal with 1:17 left in the period.
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