Bergeron won it with a wrist shot from the edge of the left face-off circle after Brock Nelson tripped Brad Marchand 40 seconds into overtime.
It was Boston's third overtime win in eight games this season. New York is 7-3.
"We hadn't had much luck in that," Bruins coach Bruce Cassidy said of overtime. "I thought it was the right call (on Nelson), he stuck his foot out and tripped him. … Good for us. They got a great overtime record, us not so much. Maybe this'll be the beginning of balancing the scales a little on that part."
WATCH: [All Bruins vs. Islanders highlights]
Tuukka Rask made 35 saves for Boston (27-8-11), which has won three in a row and nine straight games at New York (six at Barclays Center, three at Nassau Coliseum).
"I thought it was a great game tonight," Bergeron said. "It was fun to be a part of it. Even though it was a low-scoring game, there was a lot of things happening. I thought both teams were skating well and creating, but also tight-checking and not much space out there. That's what we're going to see down the road, and they're a great team. My expectations were what we just got from them. It was a great game."
Mathew Barzal had a goal and an assist for New York (27-12-4). Semyon Varlamov made 30 saves.
Barzal tied it 2-2 at 9:33 of the third period on a redirection of Josh Bailey's pass from the left circle.
"The chances were there. That's a heck of a hockey team over there," Barzal said. "Two top teams going at it. I think that's kind of the way we expected the game was going to be, kind of a chess match and a little bit of back-and-forth there, which was nice.
"That's a good defensive hockey team, similar structure to us. There's not too many breakdowns or easy goals."
