"I can't remember the last breakaway I had, actually," Rask said. "I guess it's a better thing to take them all at once than every now and then. Today they had four, which is really uncharacteristic for us. It's a good thing we got a win."
Stephen Weiss scored in the shootout to give Florida a 1-0 lead in the third round, but Bergeron kept Boston alive by beating Scott Clemmensen with a wrist shot.
After Rask stopped Mikael Samuelsson, Krejci gave Boston the victory with another wrist shot.
Bergeron had the two Boston goals in regulation, giving him seven in his last seven games. Tyler Seguin, the Bruins' leading scorer, assisted on both goals.
Julien had high praise for Bergeron after the game.
"He was by far the best player, no doubt about it … the way he competed, even his backchecking," Julien said. "He's always the first one back. You understand why at the end of the game he's exhausted because he leaves it all out on the ice. Right now we need more of those kind of guys."
Krejci was held without a point, ending his 11-game streak. It was the longest in the NHL this season.
Rask, who came in with a 3-1-0 career record and a 0.75 GAA against Florida, stopped 38 shots in regulation and overtime as he extended his career-best winning streak to seven games.
Florida had the best chances in overtime, including a power play early on. Kris Versteeg also had a breakaway in the final minute of the extra period.
The Panthers' Michal Repik had a breakaway in the final minute of the third period, but the puck went off his stick when he tried a deke to his backhand.
"It was a good hockey game," Weiss said. "It was a fun game to be a part of. We would have liked to come out on the other end of it, but if we play like that, we're going to have some success.
"We had some chances to put it away, but couldn't finish it off in overtime. Then it's 50-50 when it comes to the shootout."
In addition to Versteeg and Repik, Shawn Matthias also had two breakaways for Florida. Rask stopped him in the first period, but Matthias scored in the second to tie the game at 2-2.
"Our battle was better tonight, but when you give a team four breakaways it gives you an idea of the sharpness of our game," said Julien. "We needed more sharpness. The positive thing about that is Tuukka had a good warmup for the shootout and was ready for it."
With the victory in front of 19,018, the Bruins avoided their first losing streak since Dec. 6-8, when they came up short against Winnipeg and Florida.
Jason Garrison had the other goal for the Panthers, who picked one point to regain first place in the Southeast Division. Florida was in first for 57 consecutive days before being overtaken by Washington on Sunday.
The Panthers lost for the eighth time in 11 games (3-5-3), but turned in a much better effort than they did in their previous game, a 4-1 loss against an injury-depleted Pittsburgh that came in on a six-game losing streak.
"That was a good hockey game for us," Panthers coach Kevin Dineen said. "I liked our response to a pretty average game in here against Pittsburgh. It was definitely one of those games where we knew both teams had been kind of sputtering along. I knew that game would have a little bit of testosterone to it. It certainly had some jump out there. Unfortunately for us we weren't able to score that final goal or get that final save. That's what it came down to, the final play of the game."
Weiss had an assist on Matthias' goal was the 232nd of his career, as he surpassed Olli Jokinen to become Florida's all-time franchise leader.
Clemmensen made 33 saves.
"We were right there," Clemmensen said. "It's always tough when you lose no matter how you look at it. You can say, 'Wow, we got a point from the defending champs, a much-needed point', but at the same time, we want to win. We expect to win every game. We gave ourselves a chance to win tonight."
Already without the services of Brad Marchand, serving the fourth game of his five-game suspension, Boston also was missing center Rich Peverley, who returned home on Sunday to attend to personal matters.
Florida, meanwhile, lost veteran defenseman Ed Jovanovski to an upper-body injury in the first period. Jovanovski left the ice after being involved in a fight with Boston forward Daniel Paille and did not return.
The Miami Herald reported after the game that Jovanovski might have suffered a broken hand and will see a surgeon on Tuesday.
The fight began after Jovanovski and Paille collided near the boards in the Boston zone, and Jovanovski hit Paille in the face with his right elbow.
"I went at him and told him to watch the elbow," Paille said. "He wasn't sure what I was talking about, so I cross-checked him and he just asked me to go. I looked at the replay and I think it was an accident."
The game was a far cry from the teams' last meeting on Dec. 23, when Boston routed Florida 8-0 — the worst shutout loss in Panthers history.
"It was a hard-fought game," Bergeron said. "They're a very good team and they played hard and they were very hard on their forecheck. We seemed to find a way and that's all that matters."
Bergeron gave Boston the lead twice in the first two periods, only to have Florida tie the game.
He opened the scoring only 1:20 into the game with a seemingly harmless wrist shot from about 50 feet away that beat Clemmensen high to the stick side.
Garrison tied it at 8:28 of the second when his slap shot from the left point bounced in off Boston defender Dennis Seidenberg.
Matthias figured prominently in the other two second-period goals.
Bergeron's goal at 14:40 came on a 5-on-3 advantage, and it was Matthias' penalty for delay of game that gave Boston the two-man advantage. Bergeron scored as he stood in the crease and tipped in Seguin's pass between Clemmensen's legs.
Matthias then scored 1:14 later after he came out of the box and took a long pass from Tomas Kopecky to get a breakaway. With defenseman Andrew Ference giving chase, Matthias beat Rask with a shot through the 5-hole.
"That was a good hockey game," Matthias said. "I think we outplayed them for most of the game. That's good for the confidence. You've got to look at the positives out of that game. That's a great team over there. They've been killing teams this year. They've beaten some of the best teams 6-0. You saw what they did to us last time we were there.
"I think we outplayed them there. We had the majority of the chances. We had the puck, we had control of it. We were winning our races. Unfortunately a couple of bad bounces cost us the game. But we should hold our heads up high here."