The Bruins scored once in four chances on the powerplay, while the Flames went 0-for-5 with the man advantage.
"We just weren't quite good enough in both those areas," Gulutzan said. "We needed to be better."
With Mike Smith out of the lineup due to a lower-body injury he suffered late in Calgary's 3-2 road win over the New York Islanders on Sunday, David Rittich started in net for Calgary and did his best to keep the Flames (29-20-8) in the game by stopping 35-of-39 shots he faced.
"Ritter's been unbelievable for us this year," Bennett told Sportsnet during an interview in the first intermission. "He's stepped in when he's needed to and he's been great. We have full confidence in him back there."
Riley Nash had two goals for the Bruins (35-12-8), while Zdeno Chara scored into an empty net and David Backes had two assists.
Boston goalie Tuukka Rask made 28 saves to improve his record to 5-0-3 all-time against the Flames.
Nash tapped a pass from behind the net from Backes in behind Rittich 28 seconds into the game to give the Bruins a quick 1-0 lead.
It took the Flames just 95 seconds to respond as Garnet Hathaway forced Boston forward Matt Grzelcyk to turn over the puck behind the net and it bounced out front to Bennett, who quickly snapped a shot past Rask.
The Flames took a 2-1 lead at 9:12 of the opening frame thanks to a heads-up play by Gaudreau, who picked up the puck at the side of the net and tucked shot past Rask, who had fallen to the ice.
Late in the first, Rittich made a two nice saves to turn aside a pair of one-timers off the stick of Bergeron.
"He made some big saves for us," Gulutzan said. "It could have got ugly. I thought he kept us in there."
Boston outshot Calgary 16-8 in the second stanza and were rewarded when Nash finished off a nice three-way passing play with Tim Schaller and Charlie McAvoy to pull the Bruins into a 2-2 tie.
The Flames had a great chance to restore their one-goal lead late in the second, but Rask made a great glove grab to stop a snap shot from the slot by Mark Jankowski.
Bergeron took a breakaway pass from Torey Krug and snapped a shot just under the Rittich's blocker for a power-play goal 75 seconds into the third period.
The veteran Boston forward then added his second of the game at 9:19 when he redirected a shot by Brad Marchand past Rittich to put the Bruins up 4-2.
With four minutes left in the game, Calgary coach Glen Gulutzan opted to summon Rittich to the bench for an extra attacker, but his plan was foiled by Chara who shot the puck the length of the ice and into the empty net to seal the victory for the Bruins.
