The 21-year-old center, who was a restricted free agent, had 26 points (nine goals, 17 assists) in 70 games with the Chicago Blackhawks last season.
Dach's contract carries an average annual value of $3.625 million per season.
In a corresponding move to get under the NHL's offseason salary cap, Canadiens general manager Kent Hughes told reporters star goaltender Carey Price would be put on long-term injured reserve.
Price, 35, 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  playoff run," Hughes said. "We're not very optimistic that, short of some kind of intervention, that things will change."
Dach, meanwhile, was acquired by the Canadiens on July 7 for first- and a third-round picks at the 2022 NHL draft as part of Chicago's rebuild that also saw Alex DeBrincat dealt to the Ottawa Senators.
Dach had 59 points (19 goals, 40 assists) in 152 games with the Blackhawks after being selected No. 3 overall in 2019.