Henrik Zetterberg and Niklas Kronwall had a goal and two assists each, Daniel Cleary added a goal and an assist, and Jordin Tootoo also scored for Detroit (9-7-3), which netted three power-play goals in the second period to take the lead.
"It's one of those games where it felt like the puck was finding its way in on almost every shot," Luongo said.
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.
"Up 3-2 after the first period. After that it went sideways," Vancouver coach Alain Vigneault said.
Brunner added his second goal of the game and 10th of the season in the third before Andersson scored twice.
"This was a tough game. You're playing at 5:00 after playing a 7:00 the night before," Detroit coach Mike Babcock said, as his team earned a 4-0 win against Nashville on Saturday night. "It's one on your schedule you look at, you don't know if you'll have an opportunity to win. We feel real fortunate to have gotten back-to-back wins."
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 bad-angle shot from the right-wing boards went in off of defenseman Keith Ballard.
The Red Wings went ahead just 1:29 later when Kronwall's slap shot went in off the stick of defenseman Dan Hamhuis for Kronwall's third of the season and second in two games.
"It was nice to get a few goals. I thought the power play worked pretty good," Kronwall said. "It hasn't really changed too much to be honest with you. We just stuck with our game plan."
Daniel Sedin's second goal, with 3:42 left in the first, tied it at 2. He again teamed with Henrik, who made a perfect dump-in that caromed off the boards in the left corner of the Detroit zone to Daniel, who was cutting in off left wing. He picked up the puck and beat Howard for his seventh goal.
Higgins capitalized on Justin Abdelkader's turnover to give the Canucks a 3-2 lead with 1:34 left in the first.
Material from wire services was used in this report.