Boston won for the 10th time in 13 games and hasn't lost consecutive games in regulation since March 2012.
Florida lost for the fifth consecutive time and has dropped eight of nine.
Boston scored twice in the first to take a 2-0 lead. Zdeno Chara's scored at 3:55 when his slap shot from the left faceoff circle beat Panthers goalie Scott Clemmensen.
Bergeron connected for his seventh after converting a pass from Brad Marchand at 14:56.
Shawn Matthias scored a short-handed goal for Florida after outhustling Bruins defenceman 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.
It's not always going to be pretty, especially against a great goalie [Lundqvist] like that. My goals are never how you draw them up. It's just a matter of grinding it out and trying to create havoc in front and good things happen.
— Capitals forward Jay Beagle on scoring in their Game 3 victory against the Rangers on Monday