Tyler Seguin scored the other shootout goal and Brad Marchand had the regulation goal for the Bruins (2-0).
Boston was short-handed for 2 minutes, 17 seconds of overtime after playing the final 71 seconds of regulation short-handed.
Bergeron felt like the solid penalty killing was a simply a matter of keeping things easy.
"We've stuck to our system," he said. "I guess we've kept it simple."
Tuukka Rask made 26 saves in regulation and held the Jets to only Blake Wheeler's score in the shootout. Bryan Little, the last shooter, hit the right post to end the game.
Ondrej Pavelec made 26 saves in regulation for the Jets, but gave up scores to Boston's first two shooters in the extra session. Chris Thorburn scored for the Jets (0-2).
Seguin moved in, shifted a few times before beating Pavelec with a forehand shot to the glove side. Bergeron, the second shooter, slipped one between the goalie's pads.
The Jets played the final 1:11 of regulation and opening 49 seconds of OT on a power play when Johnny Boychuk high-sticked Little. Olli Jokinen's tip-in bid slid just wide of an open side of the net in the closing seconds.
Winnipeg also had a power play for the final 1:28 of overtime when Zdeno Chara was called for holding Wheeler, but it failed to get a decent shot on net despite being set up in Boston's end for a good portion of time.
"I think we were playing aggressive right off the bat," Bergeron said. "We didn't give them time to set up the plays they wanted because we were so aggressive."
Boston coach Claude Julien felt his team's penalty killing unit led to the win.
"That's what made the difference," he said. "I thought our PK was good. (Bergeron) and (Chris Kelly) were breaking up plays when they got over the blue line."
Winnipeg was 0 for 4 on the power play.
"It's disappointing when you get those opportunities in the end there," Jets forward Evander Kane said. "Myself, I can't be making those bad passes and not generating anything. That's something that we're going to have to get better at."
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The Bruins had a power play late in regulation when Kane was whistled for goaltender interference. Rich Peverley clanged a slap shot from the left point off the right post seconds after the ensuing faceoff. Nathan Horton had a bid in close, but Pavelec dropped to make a chest save.
Early in the third, Winnipeg's Paul Postma hit the post to Rask's right. Seconds later, the netminder, moving across the crease, made a glove save on Thorburn.
In a contest that lacked the emotion and fan enthusiasm of Boston's opening 3-1 win over the New York Rangers on Saturday, both teams seemed lackluster for long stretches in the first and second periods.
Coming of a 4-1 loss at home against Ottawa in their season opener, the Jets scored their first even strength goal of the season when Thorburn scored off the rebound of Postma's shot from the point 1:58 into the game. The assist was Postma's first NHL point.
Boston tied it 1-all when Seguin stole Jokinen's clearing pass near Winnipeg's blue line, broke in on the right wing and fed a streaking Marchand. who redirected the puck past Pavelec from the low slot with just under 6 minutes left in the opening period.
The Bruins had a few excellent scoring chances late in the second. Defenseman Aaron Johnson hit the post with just over 4 minutes to play. The Bruins best opportunity came when Seguin collected the rebound of Bergeron's shot in the front of the net, shifted to his right and missed an open net with a little over 2 minutes to play.
The Jets, who had a pair of power play chances in the second, nearly regained the lead early in the second, but Jokinen hit the side of an open net while skating on a man advantage.
Rask made a glove save of Kyle Wellwood on a clean breakaway midway into the opening period.
NOTES: Bruins D Dennis Seidenberg missed the game with an undisclosed lower body injury. Julien said after the game that he could be back on the ice tomorrow. ... It was the first of three games between the teams this season, but the only one at TD Garden. ... The teams split their four meetings last season. ... Olympic Gold Medalist Aly Raisman dropped the ceremonial first puck. ... Boston D Boychuk played his 200th career game. ... Winnipeg is in Washington, D.C against the Capitals Tuesday. ... Boston is off until Wednesday night's rematch against the New York Rangers.
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