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.
That's a really tough thing to play through; when you finally are thrown out there, the game is fast, intense, every single play matters, you've got to find a way to play through that and make smart plays. So it's a huge testament to [Vermette] and some other individuals in our locker room that are putting their own ego, their own wants and needs aside, and are ready for those opportunities when they do come.
— Blackhawks captain Jonathan Toews on teammate Antoine Vermette, who returned from being a healthy scratch in Game 3 to score the double-overtime winner in Game 4