The Bruins took a 3-0 lead in the Eastern Conference finals and one more victory will put them in position to battle for hockey's top prize. Boston is trying for its second Stanley Cup title in three seasons.
Bergeron scored the game-winning goal in 2011 when the Bruins defeated Vancouver in the seventh game of the Stanley Cup Finals and he came through again Wednesday in a tense duel that had been deadlocked since midway through the second period.
Boston will try for a series sweep when it hosts Pittsburgh in Game 4 on Friday.
It remained deadlocked until Bergeron bested Tomas Vokoun with Boston's 40th shot of the game.
Jaromir Jagr set up the winning goal by winning a battle along the boards in the neutral zone. Jagr advanced it to Brad Marchand, whose shot was deflected into the net by Bergeron.
Vokoun was given the start after being removed during Pittsburgh's 6-1 loss in Game 2.
Tuukka Rask made 53 saves for the Bruins, improving his record in this year's playoffs to 11-4. He has won four in a row and has allowed just three goals in those four games.