Ukko-Pekka Luukkonen made 18 saves for the Sabres (33-30-6), who have lost three straight and nine of 11 (2-7-2).
Buffalo remained six points behind the Pittsburgh Penguins for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"We believe we're still in it, but we've got to actually go out and play," Sabres forward Dylan Cozens said. "It's not going to be easy games. Everyone wants to win at this time of the year. It's just unacceptable by us this last stretch."
Bergeron scored 15 seconds into the first period to give the Bruins a 1-0 lead, scoring from the slot off a pass that Brad Marchand sent from behind the net through Luukkonen's legs.
"A lot of times you cannot have those good starts and you kind of play a little too loose," Boston defenseman Charlie McAvoy said. "But we wanted to make a priority to do that and not necessarily score off the shift. But it's always nice when you do that, but do it the right way, get rewarded right off the bat, and we kind of rolled from there."
Garnet Hathaway made it 2-0 at 9:01, beating Luukkonen short side from the right circle, and DeBrusk extended it to 3-0 at 11:40 off a backhand pass from Coyle near the right post.
Hampus Lindholm pushed it to 4-0 at 7:48. DeBrusk circled around the back of the net before sending a pass to Lindholm, who finished from the slot.
"We're giving too much time to players, and with all that time they get in the defensive zone they're making plays," Cozens said. "Guys are too good to have that much time and make plays off it. We need to pressure them more."