Bergeron scored his second of the game during a power play at 13:49 of the second period. Torey Krug's slap shot from the point struck Adam Pelech and caromed to Bergeron in the right face-off circle, where he scored on a one-timer to make it 3-1.
"We felt like the first period was pretty good; we were playing hard," Islanders forward Josh Bailey said. "We were getting pucks in and getting pucks on net. We're playing good teams. If the roles were reversed, we'd be the ones pushing back in the second period like they did. It's just finding a way to weather those storms a little better and try and limit their Grade A chances."
Bergeron completed the hat trick when he spun and beat Halak from the left face-off circle at 3:45 of the third period to make the score 4-1.
With his third goal, Bergeron passed Wayne Cashman for seventh on the Bruins all-time list with 278 career goals.
"No, it's not something I stop and think about, to be honest with you," Bergeron said. "I had a chance to learn from [Cashman] my first few years, and he was a great person and I've only heard good things about him as a player and the way he played the game.
"It's always great to hear and a great honor, but at the same time I'm only worried about what I can control, which is helping my team get some more wins."
Tavares scored his 25th goal when he one-timed Bailey's feed at 14:24 of the third period to make it 4-2.
Eberle gave the Islanders a 1-0 lead at 7:35 of the first period. It was his 15th goal of the season.