Toffoli took a cross-crease pass from rookie Cole Caufield and buried his fourth of the post-season on Connor Hellebuyck with a one-timer for the winner.
Erik Gustafsson and Artturi Lehkonen had the goals in regulation for Montreal, which has won seven straight post-season games after coming back from a 3-1 deficit to stun the Toronto Maple Leafs in the opening round. Toffoli, who scored 28 times in the regular season, also had an assist.
Carey Price made 14 saves for the North Division's underdog No. 4 seed in front of another boisterous crowd of 2,500 fans at the Bell Centre.
The 16th and final team to qualify for the playoffs after winning just one of their final nine regular-season games in regulation, the Canadiens are the first club to make the semifinals of the Stanley Cup tournament in 2021.
Montreal now awaits the winner between the Colorado Avalanche and Vegas Golden Knights, who are tied 2-2 with Game 5 set for Tuesday in Denver.
The Canadiens improved to 34-0 all-time when leading a series 3-0, and added to their NHL record with the franchise's 23rd sweep.
Winnipeg -- which registered a sweep of its own in the first round against Connor McDavid's Edmonton Oilers, but was minus No. 1 centre Mark Scheifele after he was suspended four games for charging on Montreal forward Jake Evans in the series opener -- got two goals from rookie defenceman Logan Stanley.
Hellebuyck stopped 39 shots.
The Canadiens were without top-4 defenceman Jeff Petry (upper-body injury) after he appeared to jam his right hand caught in one of the camera holes along the glass in Game 3. The veteran blue-liner's absence paved the way for rookie Alexander Romanov to make his NHL playoff debut on Montreal's third pair.
Montreal's seven-game playoff winning streak is the franchise's longest since Patrick Roy backstopped the Canadiens to 11 consecutive victories on the way to capturing Canada's last Cup in 1993.
The Jets, meanwhile, are now 1-6 when facing elimination since returning to the Manitoba capital in 2011.
Winnipeg, which owned a record of 6-3-0 and averaged better than three goals a game against Montreal in the regular season, found the back of the net just three times on Price after a 5-3 loss in Game 1.
Montreal opened the scoring on the power play at 8:01 of the first period on Gustafsson's first-career playoff goal, and first with Montreal since being acquired from the Philadelphia Flyers prior to the trade deadline.
After Jets forward Nate Thompson inexcusably failed to get the puck out of his own zone, the Canadiens worked it back to Gustafsson, who pump-faked a one-timer before changing the angle and firing through a screen on Hellebuyck.
Winnipeg had a couple of good looks on its first man advantage, but Price, who registered a shutout in Game 2 and allowed one against in Wednesday's 5-1 victory in Game 3, was there at every turn.
Montreal then made it 2-0 with 51 seconds left in the period when Lehkonen won a battle for positioning in front -- something Winnipeg was unable to do at either end of the ice throughout the series -- and tipped Brett Kulak's point shot for his second goal in as many games.
With their season on the line, the Jets got life from an unlikely source in the second period. Stanley, a bruising six-foot-seven rookie defenceman, took a pass and moved down to the left circle before wiring a shot upstairs on Price's shortside just 1:40 into the second.
The 23-year-old was at it again -- Stanley had one NHL goal before Monday -- at 5:29 moments after Hellebuyck made a nice save on Brendan Gallagher when he took a pass from Kyle Connor after coming off the bench and beat Price upstairs on one-timer from well out the netminder should have had to tie it 2-2.
Hellebuyck made a number of stops in the third as Montreal held a 13-1 shots edge on Winnipeg at one point before Price denied Paul Stastny on the doorstep at the other end.