Suzuki and Joel Edmondson scored in regulation for Montreal, which got its third win in four games.
Chicago collected a point, but dropped its season-high sixth straight game.
Traded from Chicago before the NHL draft, Dach beat goaltender Arvid Soderblom with a wrist shot, then held his right hand to his ear as the crowd booed.
''Nice little touch,'' Dach said of the crowd reaction, which also happened when he took the ice for the pregame warmup. ''I play the game on emotion and it just kind of came to me in the moment.''
Montreal coach Martin St. Louis understood Dach's reaction.
''To have the opportunity to be that guy that seals the deal at the end, it's good that he was in that position,'' St. Louis said. ''You want to play a big game coming back here.''
Chicago carried the play in the final minutes and in the five-minute overtime, and tied the game 2-all with 3:54 left in regulation when Taylor Raddysh scored from the crease on a rebound of Patrick Kane's power-play drive from the high slot.
Sam Montembeault made 30 saves for Montreal.
''We did a really good job of managing the puck,'' Montembeault said. ''If we didn't have anything, we just dumped it in. And guys are blocking a lot of shots. It's great for goalies.''
Soderblom stopped 21 shots and watched two Montreal shots ring off the goalposts.
''I think our game's going in a good direction,'' said Chicago defenseman Caleb Jones, whose first goal of the season tied the game.
RISE TO OCCASION
Until the dramatic shootout goal, Dach had only one shot on goal. The third-overall pick in the 2019 draft played three seasons for Chicago but never lived up to expectations, with 59 points in 152 games. He's produced with the Canadiens: four goals and 17 points in 21 games.
Montreal: Hosts the San Jose Sharks on Tuesday.
Chicago: Hosts the Winnipeg Jets on Sunday.
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