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 was definitely emotional. I really appreciated the fans. It was a cool feeling and it felt special and the ovation there at the start and then you kind of feel funny out there standing by yourself. Thinking back, I was saying just a bit ago, you think back just trying to make the NHL and then you kind of reflect on all the years being able to play for a great organization here in Calgary and all the fun I've had so far in my career. I feel very fortunate and blessed.
— Boston forward Jarome Iginla on his return to Calgary, where he played for 16 seasons