Price was injured as the third period ended in Montreal's eventual 3-2 overtime loss to the Senators on the road on Tuesday.
Peter Budaj, who will start Game 5, played the overtime period and was beaten by Ottawa forward Kyle Turris.
The Canadiens had already announced that captain Brian Gionta is out for the season due to a biceps injury, with Brandon Prust and Ryan White dealing with injuries that will prevent them playing the fifth game of the series at Bell Centre.
The silver lining for the Habs is that Budaj was arguably the best true backup goalie in the NHL this season, earning two wins against Boston to help Montreal edge the Bruins for the division title.
Budaj went 8-1-1 in 13 appearances with a 2.29 goals-against average and .908 save percentage.
Robert Mayer, who's never played in the NHL, will serve as Montreal backup on Thursday. The 22-year-old Czech netminder spent the season in the American Hockey League with the Hamilton Bulldogs.