Matthews took a feed from Patrick Marleau and beat Carey Price upstairs with 61 seconds left in the extra period.
John Tavares, with his first for Toronto, also scored, while Frederik Andersen stopped 34 shots. Nazem Kadri added two assists.
Artturi Lehkonen and Andrew Shaw scored for Montreal, which got 23 saves from Price. Max Domi – the son of former Leafs tough guy, Tie Domi – had two assists in his first game with the Canadiens.
With the Leafs viewed as legitimate Stanley Cup contenders after adding Tavares in free agency to a roster already loaded with offensive talent up front, the city has been buzzing for months at the prospect of Toronto finally ending its 51-year title drought.
The Canadiens, meanwhile, are loathe to use the word “rebuild,” but have clearly begun a transition phase following the trade of captain Max Pacioretty to the Vegas Golden Knights last month for Tomas Tatar, prospect Nick Suzuki and a 2019 second-round draft.