"The claim involving Bradley Johnson was upheld meaning his three-match suspension is withdrawn," said the FA on its website.
"The claim involving Loic Remy was dismissed therefore his three-match suspension remains with immediate effect."
The players were shown straight red cards in the 82nd minute.
Johnson will be available for Norwich's match at Cardiff City on Saturday while Remy will miss the north-east derby at home to Sunderland and two more games.
(Reporting by Rex Gowar; editing by Tony Jimenez)