Datsyuk, who netted one of Detroit's two power-play goals in the first period, deked and put a shot past Carey Price on the Red Wings' first shootout try.
Jimmy Howard, who made 16 saves, stopped Mike Cammalleri to open the tiebreaker, and Tomas Plekanec struck the post on the Canadiens' next shot. Zetterberg then clinched the win with a shot between Price's pads.
"The way I figure it is in the shootout it was my chance to sort of get a little bit of retribution," Howard said.
Brad Stuart and Datsyuk scored power-play goals 2:01 apart in the first period for Detroit, which lost 2-1 to Florida in a shootout on Friday.
"It was a tough loss for us," Zetterberg said. "I think we played well. And we came in here, we got a lot of power play from the beginning - up two goals. I think it was a well-played game, both teams coming from back-to-back games so it was good goaltending."
Cammalleri scored twice in the third for Montreal, beaten after regulation for the first time this season. The Canadiens were 8-0 in games decided beyond regulation, 4-0 in shootouts.
Price stopped 32 shots, the same number he turned aside Friday in the Canadiens' 3-2 win in Washington.
Cammalleri, who has three goals in two games, scored nine seconds into the third period -- snapping a shot past Howard to draw the Canadiens within 2-1.
He tied it at 8:45 when he slapped Andrei Kostitsyn's centering pass between Howard's legs from the edge of the crease. It was his team-leading 11th goal.
Stuart, who recorded eight of Detroit's 16 shots in the first period, scored his second of the season 12:41 in while Montreal's Georges Laraque was in the penalty box serving six minutes in minor penalties.
Datsyuk scored his sixth goal at 14:42 during a lengthy a 5-on-3 advantage after Paul Mara was called for cross-checking.
Laraque, who returned to the Canadiens lineup Friday after missing 13 games because of back pain, was called for tripping Niklas Kronwall at 8:52 following a leg-on-leg, open-ice collision with the Red Wings defenseman. A delayed penalty had already been signaled against the Montreal enforcer for his high stick on Darren Helm, which drew a double minor.
Kronwall didn't return after he fell to the ice in obvious pain following his collision with Laraque, who appeared to leave his leg in Kronwall's path.
"You know it's something that shouldn't happen in hockey," Red Wings coach Mike Babcock said. "A guy shouldn't hang his leg out there and hurt another guy. It's unacceptable and it's disappointing, obviously. We just lost a good player for a period of time. Our doctors will look at him, but he's out for a chunk for sure."
While the hit was called "bad" by Zetterberg and "dirty" by Datsyuk, Laraque doesn't expect to face any further punishment by the NHL.
"There were four refs on the ice and they didn't call anything," Laraque said. "If they called a match penalty it would be different. There was no intent. There was no reason why I would try to go and hurt him. It was a pure accident, that's why they called it tripping, so I'm not worried at all."
The Canadiens, who lost center Scott Gomez to a lower-body injury Friday, had killed Laraque's first minor penalty when Stuart put the Red Wings up 1-0 with a slap shot from the point.
Datsyuk redirected Brian Rafalski's pass from the left point past Price to make it two straight goals for Detroit's power play at 14:42.
The Red Wings, who outshot Montreal 16-3 in the opening period and 34-18 overall, enjoyed another 5-on-3 advantage for 1:27 late in the first when Canadiens defenseman Ryan O'Byrne was called for slashing while Travis Moen was in the box.
"I thought in the first 10 minutes of the game we had good momentum going," Canadiens coach Jacques Martin said. "We were playing four lines, we had good tempo, we were controlling the play. The number of penalties we took gave them the two goals, and then you've got to battle back."
The Canadiens, who will celebrate their 100th anniversary on Dec. 4, wore retro uniforms evoking the team's original sweaters. The blue throwback jerseys, featuring white trim and a letter "C" on the chest, are similar to those worn by the 1909-10 team as a charter member of the National Hockey Association.
- Stuart recorded 26 shots in his first 23 games.
- The Canadiens went 2-10 in 1909-10 during the inaugural season of the NHA, the NHL's predecessor.
- Cammalleri had 39 goals with Calgary last season.