Daniel Sedin had two goals, and Chris Higgins also had a goal for Vancouver. Henrik Sedin had two assists, and Roberto Luongo stopped 20 shots but lost in regulation for the first time this season.
Cleary tied it 3-3 at 2:57 of the second period when he tipped the puck in during a goal-mouth scramble for his fourth goal. Zetterberg's sixth gave the Red Wings the lead with 4:50 left in the second, and Brunner made it 5-3 when he calmly flipped in a rebound with 42 seconds remaining.
The Red Wings, who entered ranked 26th in the NHL on the power play, held the Canucks to only three shots in the second period.
Brunner added his second goal of the game and 10th of the season in the third before Andersson scored twice.
Daniel Sedin's first goal of the game gave the Canucks a 1-0 lead 5:45 in. He put in a one-timed shot from in front off a pass from his brother, Henrik, who was behind the net.
Tootoo tied it with 8:52 left in the first when his shot from the right-wing boards went in off of defenseman Kevin Ballard.
The Red Wings went ahead just 1:29 later when Kronwall's slap shot went in off the stick of defenseman Dan Hamhuis.