Detroit improved to 9-22-3 while Montreal fell to 15-12-6.
Tyler Bertuzzi and Mike Green (power play) scored for the Wings.
Detroit netminder Jonathan Bernier (7-9-2) made 42 saves.
Montreal goaltender Carey Price (13-10-3) had 18 stops.
Up next for the Wings is the second half of a road-home back-to-back set as they host the Los Angeles Kings Sunday night at Little Caesars Arena.
The Canadiens embark on a seven-game road trip that starts Tuesday in Vancouver.
1.Jonathan Bernier: Although a native of Laval, a suburb of Montreal, Bernier has not had a tremendous amount of success playing against his hometown team. Coming into the game, Bernier was 2-11-3 with a 3.25 goals-against average and .898 save percentage in 18 career games against the Canadiens. But Bernier, coming off a skid-snapping victory Thursday over the Winnipeg Jets, seemed to be inspired Saturday. The Canadiens, who average 34.8 shots per game, certainly kept Bernier busy. Just 12 seconds into the game, Bernier had to make a save on Artturi Lehkonen's shot. Phillip Danault had a couple of good chances in the first as well. Late in the first, Bernier made two saves during the penalty kill, including on Jordan Weal from the left circle. Early in the second, Tatar tested Bernier, who was up for the challenge. At 8:10 of the second, Bernier had a particularly good save on Nick Cousins who was at the left side of the net. Bernier even had to make a stop with three seconds remaining in the second as he saved Joel Armia's backhand. Then 17 seconds into the third, Bernier stopped Jeff Petry's shot from the right point. Then a minute later, he stoned Danault again. this time the shot coming from the right circle. In all, the Canadiens had six shots in less than three minutes of the third, including an especially nice one on Nick Suzuki on the rebound. At 16:02, Bernier made a save on Weber's blast from the left point. It was only when the Canadiens had the extra attacker that they were able to score with 46.7 seconds remaining in regulation. Tatar got his 12th goal of the season, assisted by Weber and Armia.