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 something that you don't really envision starting this way, but when you put the work in you don't feel like it's unwarranted. You definitely don't envision it unfolding this way, but you do think you can be successful at this level.
— Senators goalie Andrew Hammond, who defeated the Jets on Wednesday to improve to 6-0-1 since making his first NHL start on Feb. 18