The 21-year-old centre, who was a restricted free agent, had 26 points (nine goals, 17 assists) in 70 games with the Chicago Blackhawks in 2021-22.
Dach's contract carries an average annual value of $3.625 million per season.
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."
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.
The six-foot-four, 197-pound native of Fort Saskatchewan, Alta., had 59 points (19-40) in 152 career games with the Blackhawks after being selected third overall in 2019.
This report by The Canadian Press was first published Sept. 7, 2022.
The Canadian Press