The home side took an early lead through Portugal defender Jose Fonte only for ex-Spurs man Gary Doherty to level soon after.
Former Benfica man Fonte notched his second before the break and substitute Danny Butterfield made sure of all three points with a last-minute thunderbolt.
But Neil Warnock's men far from had it their own way, with the Canaries dominating for large periods of the encounter.
Fonte gave Palace the perfect start in the sixth minute, nodding home from close range after Paddy McCarthy's flick on.
But Palace failed to capitalise on their good start and were pegged back in the 18th minute when Doherty nodded home at the back post after David Bell's teasing free-kick.
Fonte missed a glorious chance to put his back in front in the 27th minute, but made no mistake four minutes later, rifling home, via a deflection, from 12 yards after David Marshall could only parry Sean Scannell's firm header.
However, once again the Eagles failed to assert their authority on the game and should have been punished for it in the 41st minute when on-loan Leroy Lita somehow hit the post with the goal gaping from six yards after Jon Otsemobor's low cross.
The second period didn't start well for Palace after inspirational defender McCarthy was forced off with the recurrence of a shoulder injury and replaced by Butterfield.
Eagles keeper Julian Speroni had to be alert to Mark Fotheringham's deflected drive on the hour and two minutes later he was at full stretch to prevent substitute Butterfield scoring an embarrassing own goal as the away side started the second period brightly.
Norwich continued to dominate proceedings in search of a deserved equaliser, Wes Hoolahan saw his goalbound shot blocked by Shaun Derry in the 75th minute, while Otsemobor sent his drive from 20 yards just wide moments later.
But Butterfield eased Palace's nerves in the final minute with a thunderbolt from 22 yards to seal all three points for the home side.