Rask made 29 saves and Bergeron scored two goals to help the Bruins beat the Panthers 4-1 on Thursday.
Boston won for the 10th time in 13 games and hasn't lost consecutive games in regulation since March 2012. The Bruins have 39 points, one behind Montreal and Pittsburgh.
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Boston coach Claude Julien earned his 246th victory, passing Hall of Famer Milt Schmidt for second in all-time wins with the Bruins. Julien is still behind Art Ross, who had 361.
"I don't think you can compare," Julien said. "It is two different eras and the scoring system has changed and we don't have ties now."
Florida lost for the fifth consecutive time and has dropped eight of nine.
"We are always chasing and it is tough to win games that way," Panthers defenseman Brian Campbell said. "Boston knows how to play with the lead."
Bergeron connected for his seventh after converting a pass from Brad Marchand at 14:56.
"I have learned a lot from him [Julien]," Bergeron said. "I have grown a lot as a person and a player."
Shawn Matthias scored a short-handed goal for Florida after outhustling Bruins defenseman Dougie Hamilton and recovering an errant Milan Lucic pass to beat Rask, cutting the lead to 2-1 at 3:10 of the second period.
"Just another frustrating loss," Matthias said. "Our shots weren't going in and we knew it would be a hard night."
Shawn Thornton scored at 12:43 of the third period when he knocked in his own rebound for a 3-1 lead.
Bergeron added an empty-net goal with 58 seconds remaining to cap the scoring for the Bruins.
Rask improved to 14-2-3 on the season and 6-1 all-time against Florida with a 0.99 goals-against average.
"The second period was a mess and we were pretty bad out there at times," Rask said. "I tried to back the guys up and that's my job."
Florida has allowed more goals than any team in the NHL with 105.
- Boston evened its all-time record against Florida to 15-15-4-3 and the Panthers and Minnesota Wild are the only two teams with franchise winning records in Boston.
- Clemmensen and Rask entered the game both allowing 36 goals, but Rask had recorded 600 more minutes of playing time.
- Florida has scored two goals or fewer during its five-game losing streak.
- Nathan Horton had a goal disallowed with 2:34 remaining.