Dach's deal will carry a $3.3625 million cap hit per season.
Montreal made two trades at the 2022 NHL Draft, first dealing Alexander Romanov and the 98th-overall pick to acquire New York's 13th-overall selection. Then, they packaged the 13th-overall pick with the 66th-overall pick to acquire Dach from the Chicago Blackhawks.
Dach was coming off his three-year, entry-level deal and was a restricted free agent.
The 21-year-old was drafted third overall by Chicago in the 2019 NHL Draft. Last season, Dach had nine goals and 17 assists in 70 games and 59 points in 152 career games.
Injuries have held him back from reaching his full potential with the club — a broken wrist suffered during an exhibition game ahead of the 2021 world juniors while captaining Team Canada saw him unable to build on his solid rookie season, suiting up for just 18 games in the COVID-shortened 2020-21 NHL campaign.
In a corresponding move to get under the NHL's off-season salary cap, Canadiens general manager Kent Hughes told reporters star goaltender Carey Price would be put on long-term injured reserve.
The 35-year-old played just five games last season because of a knee injury after leading Montreal to a surprise berth in the 2021 Stanley Cup final.
Price, who won last season's Bill Masterton Trophy as the player who best exemplifies the qualities of perseverance, sportsmanship and dedication, is heading into the fifth season of an eight-year deal that carries an annual cap hit of $10.5 million.
"Carey's knee has not responded to any kind of treatment and rehab that's been attempted since he had the surgery after the (2021) playoff run," Hughes said. "We're not very optimistic that, short of some kind of intervention, that things will change."
With files from The Canadian Press