Montreal scored the game's first three goals and then had to come from behind to force the overtime.
The teams combined for five goals in the opening period, at the end of which the Canadiens held a 3-2 lead.
Tomas Plekanec, Max Pacioretty and David Desharnais all scored in the first 12 minutes to seemingly put Montreal in charge, but the Senators closed within one by the end of the period on goals by Clarke MacArthur and Erik Karlsson.
Montreal bounced back with the only goal of the third, that coming from Plekanec with 8:58 to go in regulation.
Price made 40 saves to collect the win. Ottawa's Craig Anderson let in five of the 23 shots he faced.