The Eagles went into the game with only one win from their previous 10 games and were behind just before the break when Grant Holt got on the end of Simon Lappin's cross.
However Palace were a different side after half-time and levelled through Julian Bennett before Anthony Gardner's header completed the turnaround.
Norwich pressed for an equaliser but Palace held firm to register a win that meant they were only still in the relegation zone by the virtue of goal difference.
The Canaries came flying out of the traps and came agonisingly close to taking the lead after just seven minutes.
Palace's Alex Marrow lost the ball in midfield and the lively Wes Hoolahan played in Simeon Jackson who saw his shot cannon off the crossbar.
Moments later Jackson almost got in again after a poor backpass but Crystal Palace goalkeeper Julian Speroni was quick off his line to clear.
The visitors responded well to the early pressure and almost took the lead themselves when Edgar Davids' cross fell into the path of Owen Garvan who flashed a shot just wide of the post.
With 15 minutes gone Norwich were awarded a free-kick and Lappin flighted the ball to Leon Barnett who saw his header saved by Speroni.
However Palace continued to look dangerous on the counter attack and Andrew Dorman sent Wilfred Zaha through but his shot went just wide.
Zaha was looking dangerous for the Eagles and he got past a couple of challenges before being denied by a superb last-ditch tackle from Andrew Crofts.
Norwich almost went in front when Lappin's corner was cleared back to him and his shot was hacked off the line by Davids.
Two minutes before the break Hoolahan played the ball wide to Lappin who crossed for Holt and he made no mistake from close range to put Norwich in front.
Crystal Palace boss George Burley brought Kieran Djalili on for Davids soon after the break and the visitors immediately looked a more dangerous attacking threat.
With 56 minutes gone they were level when Zaha played a short corner to Garvan and his cross found Bennett who buried a header home from six yards.
Norwich goalkeeper John Ruddy made a good save from Dorman to keep the scores level but Palace went in front after 63 minutes.
Garvan played a corner into the near post and Gardner got in front of his marker to head home.
The home side tried to get back into the game and Anthony McNamee flashed a shot just over the bar as Palace held on for a vital win.