The two clubs were charged with failing to ensure their players conducted themselves in an orderly fashion and/or refrained from provocative behaviour in or around the 82nd minute of a goalless Barclays Premier League draw at Carrow Road on January 28.
An FA statement said: "Newcastle admitted the charge and accepted the standard penalty of £20,000, which includes a third-off for admitting.
"Norwich, who had a similar breach earlier this season, denied the charge, which was subsequently found proven, and were fined £30,000.
"Both clubs were warned as to their future conduct."
The incident happened after Magpies striker Loic Remy and Canaries' midfielder Bradley Johnson clashed off the ball, with referee Chris Foy showing both players a red card.
Both clubs appealed against the dismissals, but while Johnson's claim was upheld, Remy's was rejected and he will serve the second leg of his three-game ban at Chelsea on Saturday.