Eberle tied it 1-1 with a power-play goal at 16:40 of the second. Eberle took a pass from Mathew Barzal on the goal line to the right of the net, pivoted to the front and beat Rask to the far post high.
The Islanders thought they took the lead when Cal Clutterbuck scored at 4:52 of the third period, but the goal was overturned when replays concluded New York entered the Bruins zone offside.
After Cehlarik gave them the lead, the Bruins killed a double minor for tripping and slashing against defenseman Matt Grzelcyk.
The Islanders went 1-for-5 on the power play.
"I thought we moved the puck well on the power play; that wasn't really the issue today," Eberle said. "They're a good team over there. They don't give you much. We definitely had our opportunities, though."
Bergeron, who is the 334th NHL player to play 1,000 games, scored an empty-net goal at 19:05 for the 3-1 final. Pastrnak had a chance to score into the empty net but dropped off the puck to Bergeron on his special night.
"It means a lot, obviously," Bergeron said. "I was surprised. I saw him look a couple times, I was like, 'Oh, he's going to drop it.' But we were definitely alone, and I very much appreciate it."