Tim Schaller's goal that made it 4-1 at 7:25 of the second period chased Rangers goaltender Henrik Lundqvist and brought boos from the crowd at Madison Square Garden.
After Rick Nash gave the Rangers an early lead with his 16th goal five minutes into the first period, the Bruins began their relentless barrage.
But the Bruins forward recovered his rebound, then sent the puck in front to Nash as Rangers defenseman Tony DeAngelo was caught out of position.
The Bruins extended the lead to 3-1 at 3:47 of the second when Bergeron's long shot from just inside the blue line skittered past Lundqvist after the goaltender seemed to make a pad save.
Schaller extended the lead with his eighth of the season before Bergeron scored his second of the game - this time shorthanded - at the 10-minute mark.
The goal was Bergeron's team-leading 24th and came with assists to Chara and Brad Marchand, who returned after serving a five-game suspension for elbowing New Jersey's Marcus Johansson.
NOTES: Lundqvist made his 788th career appearance, tying Gilles Meloche for 17th place on the NHL goalies list.
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