Curtis McElhinney made 19 saves for the Hurricanes, who defeated the Washington Capitals in seven games in the first round before sweeping the New York Islanders in the second round.
"Did I think we'd be here? Maybe not, but we were going to see if we could get here," Carolina captain Justin Williams said. "We're good enough to be in the final four, and that means we're good enough to do it again next year and get better and learn from this. Right now, it hurts, and it will for a little while."
Pastrnak gave the Bruins a 1-0 lead with a power-play goal at 4:46 of the second period. He beat defenseman Calvin de Haan to the net and tapped in a centering pass from Brad Marchand off the rush.
Bergeron finished a give-and-go with Pastrnak from the slot on the power play with 1:26 remaining in the second to make it 2-0. Bergeron has six power-play goals in the Stanley Cup Playoffs.
The Bruins went 7-for-15 on the power play in the series. The Hurricanes were 1-for-14.
"Everything's going OK, then we get into [special teams] where it's obviously a huge advantage for them, and they made us pay," Carolina coach Rod Brind'Amour said. "You've got to give them credit. There's just no weakness, no matter who they throw out there."