The London 2012 Olympic bronze medallist and Canaries supporter was at the Emirates Stadium to watch his side get knocked out of the competition after losing 2-1 in extra time.
Ogogo, who grew up in Lowestoft and had a spell in Norwich's youth squad, says that he was targeted by a fellow Norwich supporter in the stands after being mistaken for an Arsenal fan.
"Terrific support tonight by Norwich fans, nine thousand strong," the 28-year-old tweeted. "Although I did get racially abused for cheering. In this day and age it's disgusting.
"Worst of all it was by a Norwich fan that thought I was Arsenal. Utterly disgusted by that behaviour. It shows how backwards some people are."
Norwich have been in touch with Ogogo and will launch an investigation into the incident.
A statement from the Championship club read: "We are very concerned by reports from Anthony Ogogo that he suffered racial abuse from a supporter in the Norwich section.
"As a club we have a zero-tolerance policy on racism and all other forms of discrimination and we have contacted Anthony this morning.
"As always, any supporter found to be guilty of discriminatory abuse will be banned indefinitely by Norwich City and face possible criminal investigation."
Ogogo is currently preparing to make a comeback in the ring after suffering a fractured eye socket in his loss to Craig Cunningham.