Nathan McSeveney, 20, from Cumnock in Ayrshire, was taken to Glasgow Royal Infirmary from Exit 33 of the stadium.
Police Scotland said his relatives had been informed along with the Health and Safety Executive.
It is understood he fell down the exit vomitory when leaving the ground.
A force spokesman said: "There are no suspicious circumstances. A report will be prepared for the procurator fiscal."
A statement posted on the Celtic club website said: "Everyone at Celtic Football Club is deeply saddened to learn of the death of a young supporter following last night's match between Scotland and the Republic of Ireland.
"This was a terrible, tragic incident and our sincere thoughts and prayers are with the young man's family at this time."
A crowd of 55,000 saw Scotland win the match 1-0 thanks to a Shaun Maloney goal.