The promoted Canaries, who had hit 26 goals without reply in their three previous matches against summer cannon fodder, found Championship side Palace a tougher proposition at Selhurst Park.
And young substitute Nathaniel Pinney secured a win for the hosts when he rounded John Ruddy and slid the ball into an empty net a minute from time.
Palace had also created the best chance of the first half when Leon Barnett had to hack a Darren Ambrose shot off the goal-line.
Earlier, Eagles skipper Paddy McCarthy denied Simeon Jackson with a last-ditch challenge and Bradley Johnson headed Grant Holt's cross wide.
City manager Paul Lambert made 10 changes in the second half but his most recent signing, Tottenham loanee Kyle Naughton, was not in the squad.
Palace are weighing up a loan move for Stoke defender Andrew Davies, who played the first 45 minutes.