Substitute David Strihavka scored the only goal of the game with a well-taken header after Dion Dublin had headed Simon Lappin's right-wing corner back along the six-yard line 15 minutes from time.
The Canaries kicked off attacking the Barclay End goal, but in the fourth minute David Marshall had to be at his best to save Stuart Green's free-kick, which found its way through City's defensive wall.
Tom Soares had a shot blocked by Dublin, while at the other end Jamie Cureton connected with Jon Otsemobor's low right-wing cross, but his shot curled wide of the far post.
The referee angered the home fans by penalising their hero, Darren Huckerby, when they thought the free-kick should have gone the other way.
The Norwich winger seemed to be short of match fitness and failed to make the most of several crossing opportunities.
Even so, Huckerby's pace was too much for Green, who was booked for body-checking him.
Four minutes before half-time, Otsemobor put in another low cross from the right, but Darel Russell's shot hit the top of the crossbar.
In the first minute of the second half, Cureton collected Huckerby's cross with his back to goal, but his shot on the turn flew well wide.
Chris Brown fired Luke Chadwick's cross over the bar before making way for match-winner Strihavka, while Palace's Dougie Freedman replaced Besian Idrizaj.
Paul Dickov was booked for a late challenge on Otsemobor and was promptly replaced by Clinton Morrison.
Lee Croft replaced Chadwick and within seconds had won the corner which led to the goal.
Lappin's right-wing flag kick was headed back across goal by Dublin and Strihavka slipped his marker and found the back of the net with a firm header.
Cureton fired into the side-netting and, in the 90th minute, he was foiled by Julian Speroni when sent clean through.
But in the end it made no difference as Norwich collected a much-needed three points as they leapfrogged Palace in the Championship table.