Jeff Petry had two assists, and Sam Montembeault made 37 saves for Montreal (20-49-11), which lost 6-4 at the Ottawa Senators on Saturday and has been eliminated from playoff contention.
It was Montreal's first home game since Canadiens legend Guy Lafleur died at the age of 70 on Friday. Hockey Hall of Famers Yvan Cournoyer, Guy Lapointe and Bob Gainey were among Lafleur's former Canadiens teammates on hand for a pregame tribute that featured an ovation lasting more than nine minutes.
After the game, the Montreal players gathered in their end of the rink and raised their sticks to salute Lafleur's banner for his retired No. 10 hanging in the rafters.
"I think a few of the guys decided to do that," Canadiens forward Nick Suzuki said. "It's a great tribute. We did everything we could tonight to get the win, we just came up short. You could tell in that third period we were doing everything possible to score and get the win for him. So just to give him a salute up in the rafters, very special."
Bergeron gave Boston a 1-0 lead at 15:03 of the first period on the rebound of Jake DeBrusk's shot from in front.
Haula made it 2-0 at 18:03 after Mike Hoffman slashed his stick on a breakaway. Haula overskated the puck at center ice on the penalty shot but did not touch it before gathering it and scoring on a low shot.
"I think we were all laughing," Swayman said. "Good for him, he faked everyone out in the building, so it worked out."