The Blues midfielder was dismissed by referee Steve Bennett minutes into the defeat at Carrow Road for hitting Canaries defender Thomas Helveg.
Johnson will have to serve an automatic three game ban for that offence plus an extra one game as it is his second sending off of the season (he was shown a red-card in the home game against Charlton earlier this season).
The Northern Ireland international will therefore miss the final game of the season against Arsenal at St Andrew's on Sunday and the first three Premier games of next season.