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.
[He's] real confident with the puck now, getting it off his stick quick and no second-guessing. We need that. He's such a good guy in the room. He works so hard. That's the big thing. For not a big man, he just fights for every puck and when he scores, the guys appreciate that even more.
— Winnipeg Jets coach Paul Maurice on Mathieu Perreault, who scored two goals in win against Chicago Blackhawks on Tuesday