MacKenzie scored in regulation for Florida, and Luongo made 23 saves.
Troy Brouwer scored in regulation for Washington and Holtby stopped 28 shots. The Capitals have points in six straight games.
"Even though we lost, it was very entertaining," Brouwer said. "We were having a little fun on the bench. We would have liked to come out on the other end of it. Anytime you go 20 shooters in a shootout, that's entertaining."
The Capitals are 1-3 in shootouts this season.
"At first you're (thinking), `Just get the shootout victory,' but once it got to 15, 16, 17, actually it was quite fun," Capitals coach Barry Trotz said. "You just sort of lived in the moment and hope that you got the two points. Unfortunately, we didn't."
But not everyone enjoyed the extended shootout, which lasted 17 minutes, 43 seconds.
"It definitely wasn't fun to be a part of it," Holtby said. "Being on the end that I had to make a save numerous times when the game was on the line and I didn't; that's not a fun spot to be in."
The Panthers are 4-0-1 in their last five home games. The win was only their second in their past 12 games against the Capitals. The Panthers lost Oct. 18 in a shootout at Washington.
The previous record for the Panthers was 11 shootout rounds on Nov. 28, 2007, also against the Capitals.
Florida is 3-4 in shootouts.
"That was exciting. Everybody loved that, especially with the way it ended, winning a game like that with Bjugstad's goal," Panthers coach Gerard Gallant said.
With the score tied 1-all at 3:08 of the third, it appeared the Capitals took the lead when the puck slid by Luongo and into the net. The goal was overruled after a video review because it was determined that Laich kicked in the puck with his left skate.
The Panthers took a 1-0 lead when MacKenzie one-timed a rebound at 9:36 of the first. The Capitals tried to clear from behind the net and the puck went off the skate of Florida's Scottie Upshall and out in front, where MacKenzie beat Holtby on the glove side.
The Capitals tied it on Brouwer's power-play goal in the second. Alex Ovechkin passed from the left circle across to Brouwer in the slot. His wrist shot beat Luongo on the glove side at 14:35 for his ninth goal of the season.
"I think we made some history tonight, and the chances of that happening again are one in a million," Trotz said.
NOTES: Holtby made his 13th consecutive start for the Capitals. ... Panthers defenseman Dylan Olsen played his 100th NHL game. ... Upshall and Aleksander Barkov returned to the Florida lineup after missing nine and six games, respectively.