Vincent Trocheck scored, and Jaroslav Halak made 19 saves for the Rangers (33-16-9), who have lost three in a row (0-2-1) after a seven-game winning streak.
New York had its road point streak end at 10 (6-0-4), one shy of its record set in 1939-40.
"In this league, we don't have any time to feel sorry for ourselves," Rangers forward Jimmy Vesey said. "We've got a game against Washington on Saturday, and that's a tough team with a lot of skill. We have to be ready to play."
The Red Wings took a 1-0 lead at 15:33 of the first period when Copp tipped in Rasmussen's pass from the edge of the crease.
Trocheck tied it 1-1 at 6:06 of the second period, taking advantage of a Robert Hagg turnover to score from the slot.
Dylan Larkin stole the puck at the Rangers blue line and set up Filip Zadina to make it 2-1 at 8:18. Larkin has 14 points (seven goals, seven assists) in his past nine games.
"That was a huge goal because we had started to get our game back after the long trip, and all of a sudden we blow up and the puck is in our net," Red Wings coach Derek Lalonde said. "We needed that goal to get the momentum back and [Zadina] stepped up."
New York nearly tied it at 17:09, but Husso kicked out his left skate to rob Adam Fox.
"I saw him coming, so I just tried to move in that direction," Husso said. "It was a very lucky save."