Filip Zadina scored both goals for the Red Wings (7-16-3), who have lost three in a row. Jonathan Bernier made 27 saves.
"We played good enough to win without a shadow of a doubt," Detroit coach Jeff Blashill said. "They had their chances too. We won more puck battles. We weren't perfect. We can't give up those two goals in the third, but a way better effort for us for sure. We've got to build on it."
Svechnikov played against his older brother, Evgeny Svechnikov, for the first time in their NHL careers. Evgeny had an assist on Zadina's second goal.
"I was a little nervous my first couple shifts, but after a couple shifts, I started thinking about my game," Andrei Svechnikov said. "I started focusing, and it was good after that."
Brind'Amour earned his 100th NHL win in his third season with Carolina.
"I did think it would be this stress because I was an assistant coach (for seven years)," Brind'Amour said. "But man, it's not good for you, I don't think, this coaching thing. Can't be healthy."
The Hurricanes broke a tie with three goals in the third period.
Staal gave Carolina a 3-2 lead 45 seconds into the period on a long rebound in the slot. Svechnikov slipped a pass across the slot to Fast for a backdoor tap-in to make it 4-2 at 2:50. Niederreiter scored an empty-net goal at 14:55 for the 5-2 final.
"We started to create a forecheck with our speed and started creating turnovers quickly," Staal said. "We just did a better job of that in the third."