Price stopped 25 shots overall, including 11 shots in the opening period. He got his first playoff shutout in a 1-0 win in Game 4.
Mike Komisarek opened the scoring 3:31 in amid one of the first of countless roars from the raucous Bell Centre crowd over the course of the evening.
The Canadiens took control of the game in the second, outshooting the Bruins 17-6 while building a three-goal lead on a superb goal by Mark Streit midway through the period and Kostitsyn's second goal of the series at 15:13.
Kostitsyn punctuated the win with his second of the game, a power-play goal with 2:02 remaining in the third. Sergei Kostitsyn scored with 7.3 seconds remaining.
Top-seeded Montreal will face Philadelphia in the second round, if the Flyers win their Eastern Conference quarterfinal tonight in Washington. Otherwise, the Canadiens will face the New York Rangers.
Tim Thomas stopped 30 shots for Boston, which fell short in its bid to overcome both 2-0 and 3-1 series deficits for the first time in team history.
CAPITALS 4, FLYERS 2
In Philadelphia, held without a goal for four consecutive games, Alex Ovechkin scored twice in the third period to lift Philadelphia to a win over Washington and send the series back to D.C. for a decisive Game 7.
The Capitals rallied back from a 3-1 hole and have a chance to win only the second Game 7 in franchise history tonight.
The other one should be painfully familiar to the Flyers faithful: Dale Hunter scored an overtime goal that led Washington past Philadelphia 20 years ago this month.
The Caps are trying to become the 21st team in league history to come back from a 3-1 deficit and win a best-of-seven series.