Jose Theodore stopped 24 shots for the Panthers, whose last 11 home games have been decided by one goal.
Olli Jokinen and Rene Bourque scored for Calgary, and Leland Irving made 39 saves in his NHL debut, but the Flames dropped their third straight game. Irving, a 2006 first-round pick, was recalled from the minors on Dec. 5.
The Panthers had lost their previous four shootouts, and had just one goal in 14 attempts before Friday. Florida's only other shootout victory this season was on Oct. 15 in the home opener against Tampa Bay.
"It's nice to get the monkey off our back with the shootout," Weiss said. "There's always confidence. If you lose that, you're lost. We knew we were going to turn the page on that. It was just a matter of time. Theodore keeps us in every one."
"I don't know, maybe a little bit. But I wanted it, too," Kulikov said. "You watch us losing in a shootout a few games in a row. I wanted it and it turned out good for me."
The Flames couldn't take advantage of a well-played game by Irving, who looked at times like a veteran.
"After those first two shots, the adrenaline kind of takes over," Irving said. "It's all kind of a blur right now.
"It would have been to get the two points, obviously. That's going to stick in my mind for a bit."
Irving's play also drew praise from the Panthers.
"He played a heck of a game," Florida coach Kevin Dineen said. "You like to see a guy get that first NHL game. That's always a special moment. Certainly as much as you enjoy seeing that, you sure like the two points."
Bergenheim's second goal made it 2-2 with 4:35 left in regulation. Shawn Matthias took a shot on net that was blocked by Irving, but Bergenheim wristed in the rebound.
"Every time you're down and you get a big goal like that, it's a great feeling," Bergenheim said.
The Flames took a 2-1 lead on Bourque's power-play goal 3:55 into the third. Jokinen passed to Bourque, who was waiting at the right corner of the crease. His wrister beat Theodore on the short side.
Florida took a 1-0 lead on Bergenheim's first goal 6:57 in. Irving stopped Matthias' initial shot, and Bergenheim was there to poke in the rebound.
Calgary tied it at 10:00 when Jokinen backhanded a rebound of Curtis Glencross' shot from in front that Theodore couldn't control.
The Flames had a 5-on-3 power play for 1:32 of the second, but couldn't take advantage.
"In the last eight minutes in the game, I thought we looked a little tired," Flames coach Brent Sutter said.
Florida RW Matt Bradley played in his 650th NHL game. ... Florida D Erik Gudbranson missed his second straight game because of a lower body injury. ... Jokinen has five goals and six assists in eight games. ... Irving was 15-7 with a 2.30 goals-against average for AHL Abbotsford.
Copyright by STATS LLC and The Associated Press
SUNRISE, FL - DECEMBER 16: Scott Hannan #23 of the Calgary Flames stretches along the boards prior to the game...