Goaltender Tuukka Rask made an astonishing 53 saves to keep the Bruins in the contest before Bergeron's tip-in at 15:19 in the second overtime period earned Boston a third straight win in the series.
Boston now leads the best-of-seven affair 3-0 and have a first opportunity to book a Stanley Cup Finals berth in Game Four at home on Friday.
The top-seeded and favored Penguins have been second best in the series so far, scoring just two goals in three games with the threat of elimination looming large on the horizon.
Pittsburgh has not been swept out of the playoffs since 1979.
Their latest disappointment was not the result of a lack of effort, with goaltender Tomas Vokoun stopping 38 shots one game after he was pulled early.
David Krejci put the Bruins ahead just 1:42 into the game, whipping the Boston crowd into a frenzy with his NHL-leading ninth goal of the post-season.
Chris Kunitz's snap-shot tied the game at 1-1 in the second and the teams remained deadlocked through the third period and opening overtime session.
The Penguins piled intense pressure on Rask throughout the third period and both overtimes but could not find a decisive breakthrough.
Penguins captain Sidney Crosby remains without a point for the series against a Bruins team that has successfully stifled the potent Pittsburgh offense.
(Writing by Jahmal Corner in Los Angeles; Editing by John O'Brien)