Wearing the captain's 'C' for the first time, Crosby scored late in the second, and assisted on Ryan Malone's power-play goal 14:50 into the third as Pittsburgh drew even twice to force overtime.
Crosby received Maxime Talbot's behind-the-net pass to the right of the goal and banked a shot off Komisarek and past goalie Yann Denis 17:04 into the second to tie it at 1.
Kovalev added two assists and Andrei Kostitsyn also had one of each for Montreal. Kostitsyn, whose brother Sergei made his Canadiens debut, scored during a 4-on-3 advantage to give the Canadiens a 1-0 lead 6:41 into the second period.
He added an assist as Corey Locke scored 11:43 into the third period to restore Montreal's one-goal lead at 2-1.
Fleury made 19 saves.
Montreal, which hosts Pittsburgh again Tuesday night, played the final two periods with four defensemen after losing two prospects to injury.
Andrew Archer was taken to hospital for X-rays as a result of facial injuries sustained in a first-period fight with Penguins enforcer Georges Laraque. Archer had a cut under his left eye and was bleeding from his nose when he left the ice to go to the dressing room after the brief and one-sided fight 14:07 in.
Defenseman Mathieu Carle injured a knee later in the first and didn't return.
Huet stopped all 12 shots he faced, including nine in the first period, before he was replaced by Denis 14:49 into the second.
Top Pittsburgh draft pick Angelo Esposito made his Penguins debut. Esposito, a Montreal native chosen 20th in the June draft, played on a line with Talbot and Adam Hall.