Phillip Danault scored, and Carey Price made 20 saves for Montreal (11-9-5), which has lost five straight (0-3-2) and has scored two goals or fewer in four straight.
"It's tough right now, but there were a lot of positives from the game tonight," Canadiens coach Claude Julien said. "Unfortunately, we're leaving precious points behind us."
Weber, who had an assist, was playing his first game since Dec. 16, 2017. He had arthroscopic surgery June 19 to repair a torn meniscus in his right knee, which followed surgery March 13 to repair a torn tendon in his left foot.
"It definitely felt better as the game went on," said Weber, who was named Canadiens captain Oct. 1. "I think it's going to keep getting better as we go here, get more comfortable being in those game situations. Obviously, practice is different than being in those actual situations where you have to read and react and things are happening quickly. It will be good to go along here and get better as we go."
Rask gave the Hurricanes a 1-0 lead at 7:45 of the first period when he redirected Jaccob Slavin's point shot past Price. It was his first goal in four games since he returned after having surgery on his right hand Sept. 13.