Artturi Lehkonen scored twice, and Carey Price made 34 saves for Montreal (18-17-7), which is 0-4-1 in its past five games.
"Obviously, it's not fun, no one wants to lose, but we have a really tight-knit group in here," Canadiens forward Max Domi said. "And we're battling some adversity right now, we have for parts of this year and we've overcome it. But at this point of the year you can't let it drag on too much, so we've got to find ways to get those two points. At the end of the day we got one point tonight, it's not what we wanted but it's better than not getting any."
Lehkonen gave the Canadiens a 1-0 lead at 8:42 of the first period. Domi got his 20th assist to extend his point streak to nine games (six goals, six assists).
Zach Aston-Reese scored 46 seconds later to tie it 1-1.
Lehkonen gave Montreal a 2-1 lead at 3:37 of the second period. It was his 100th NHL point.
Bryan Rust tied it 2-2 at 5:24 of the third period.
Lehkonen had an opportunity to put the Canadiens ahead at 12:15 of the third, but Murray made the save on his backhand shot on a breakaway.
"It was kind of a broken play," Murray said. "He came off the wall and got a little saucer pass to the middle, so I had time to get out and get set. He made a nice move and I was lucky to get a pad on it."