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.
There's not much to say. We played like [garbage]. I think every guy in the room should be completely embarrassed about how they played tonight. Every single person. To start a road trip, one of the biggest road trips and have a division rival chasing you down and have a ton on the line, it's ridiculous how we played tonight.
— Anaheim forward Andrew Cogliano after the Ducks' 7-2 loss to the Flames on Wednesday