The Canaries received a fine of £30,000, while Newcastle's amounted to £20,000 after both clubs accepted an FA charge of misconduct on Monday.
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While Johnson had his ban overturned in a subsequent appeal, Remy's ban was upheld, but both clubs have been warned about their future conduct.
A statement from the Football Association read: "Norwich City and Newcastle United have been fined £30,000 and £20,000 respectively following an FA charge of misconduct.
"The charge was that both clubs failed to ensure their players conducted themselves in an orderly fashion and/or refrained from provocative behaviour in or around the 82nd minute of their fixture on 28 January 2014.
"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."