Magnus Hellberg made 27 saves for the Red Wings (21-19-8), who played their last game before their bye week. They return Feb. 7, when they'll host the Edmonton Oilers.
"This is exactly what we could have asked for," Detroit coach Derek Lalonde said. "We gave ourselves a chance. Probably our best push was the end of the second, and those were our best looks. We just couldn't get one past [Sorokin] then. And they were OK in the third, but obviously not enough to get one behind him."
Lee gave New York a 1-0 lead at 4:44 of the second period. Nelson sent a feed from the right face-off circle that deflected off Palmieri and Lee and got between Hellberg's skates.
Nelson made it 2-0 at 4:31 of the third period with a wrist shot from beyond the circles that found its way through traffic.
"We were moving our feet," Palmieri said. "We did a good job. They had a couple of chances in the offensive zone, but it really wasn't much. So, we just did a good job of breaking pucks out, getting up ice and getting on the forecheck. That's kind of our bread and butter as a team, and creating offense off that forecheck is how we're successful. We were able to do that tonight."