The Bruins will face either the San Jose Sharks or the St. Louis Blues in the final. The Sharks lead the best-of-seven Western Conference final 2-1.
With the Hurricanes needing a miracle to overturn a 3-0 series deficit, David Pastrnak shot Boston into the lead in the second period to silence the Hurricanes fans at Raleigh's PNC Arena.
Bergeron then grabbed his first of the night to make it 2-0 before firing the Bruins into a 3-0 lead in the third period.
Brad Marchand completed the scoring for Boston, who will now look to bring yet another professional sports championship back to the city following the New England Patriots' Super Bowl win in February and the Boston Red Sox's World Series triumph last year.
Hurricanes goaltender Curtis McElhinney made 19 saves as Carolina's battling playoff campaign came to an end.
The Hurricanes had earlier eliminated the defending champion Washington Capitals in seven games in the first round before sweeping the New York Islanders.