Anthony Richard scored his first NHL goal, and Allen made 34 saves for Montreal (15-15-3), which has lost four of five (1-3-1).
"They played on top of us tonight," Canadiens coach Martin St. Louis said. "We really struggled versus their pressure. Part of that is them, part of it is us not having a lot of gas and being tired. It's tough to manage that kind of pressure when you just don't have the legs, you don't have the lungs. The thing that goes first is the mind, and it's hard to make good decisions when the mind's gone.
"I was really pleased to get to the overtime. If we got two points, we would have stole two points. We didn't deserve two tonight, and I'm happy we got the point."
Richard scored on a breakaway to make it 1-0 at 1:48 of the first period, skating past two defenders in the neutral zone before beating Georgiev glove side. Richard played 389 games over eight seasons in the American Hockey League prior to scoring in his fourth NHL game.
"I really like that play when I've got a [defenseman] on my back when coming with a lot of speed," Richard said. "That goalie, they're kind of cheating on their blocker side usually because I'm coming with so much speed. They kind of know you're trying to go blocker side, so sometimes when you just -- a little bit like [Nick] Suzuki is doing in the shootout -- if you can find a little hole on the opposite side, sometimes you can get lucky and it goes in.
"It's a nice moment for me and my family. So, even with the loss, I'll try to enjoy it a little bit, and it should be a nice couple hours to text my family and my friends."