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 really exciting. I'm pretty sure that when I play my first game I'm going to be emotional. To be back on the ice playing a game, being in game situations, with all the routines and rituals I do before games and during the game, I feel like I'm going to be emotional. I'm going to be really happy.
— Montreal Canadiens forward Tim Bozon on playing for the first time since his life-threaning illness