Hill, acquired from the Arizona Coyotes in a July 17 trade, made 20 saves in a 4-3 season-opening win against the Winnipeg Jets on Saturday.
"He's come up with some big saves," Sharks coach Bob Boughner said. "And what I love about it, he's into the game, on the bench and TV timeouts, he's worried about getting everybody else going. He's a leader in his own right."
Jake Allen made 20 saves for Montreal (0-4-0), which has scored three goals during its longest losing streak to begin a season since losing its first five games in 1995-96.
"I think every guy in here knows that this is not good enough," Canadiens defenseman Jeff Petry said. "So if it comes down to [having a players-only meeting] and we have to do that, we shouldn't be in that situation four games into the year."
Timo Meier and Erik Karlsson each had a goal and two assists, and Logan Couture had two assists.
Dahlen gave San Jose a 1-0 lead at 1:22 of the first period. He and Meier each redirected Brent Burns' wrist shot from the right point past Allen on the Sharks' first shot on goal.
Dahlen, the son of former Sharks forward Ulf Dahlen, made it 2-0 at 3:36 when he scored on a rebound of Couture's shot.
"I actually gave him a call after the game," Dahlen said. "I've been calling him after every game almost since I was, I don't know, 10? So I gave him a call after this one too and it was pretty fun. He was just happy. He thought I played well, and so did I."