Staffordshire Police said the man, thought to be a Blackburn fan, was taken to hospital in a "critical" condition.
Chief Inspector Adrian Roberts said: "At the commencement of the second half shortly after 4pm, I was made aware of an incident in the concourse area of the south stand, which housed the Blackburn away supporters, of a male who was in a collapsed and unconscious state.
"As a consequence of that, club stewards and police officers went to attend that incident and confirmed that the male was in an unconscious state and required medical support urgently.
"He was treated initially in the concourse and has been taken to the University Hospital of North Staffordshire, where the update I have at the moment from the ambulance service is that he is still unconscious and in a critical situation."
He said they were still waiting for further updates on the man's condition.
It is not known where the supporter, who is described by police as white and in his late-20s to early-30s, is from but police said he is believed to be a Blackburn fan because of the area he was in.
Ch Insp Roberts said an investigation had been started into how the man came by his injuries and a man in his 30s had been arrested.
He said: "It has been confined to the concourse in the Blackburn area and during the second half we contained and preserved that area and we have got an ongoing investigation at the moment to ascertain how he has come by his injuries."
He said there were just over 1,300 Blackburn fans at the game, and confirmed no Stoke fans were involved. He said officers were speaking to witnesses and examining CCTV to find out what had happened.
The match saw Stoke beat Blackburn 3-0.
A message on the Blackburn Rovers website said a man was in a critical condition following the game.
It said: "The man was among a 1,300-strong Rovers following in the South Stand, occupied by visiting supporters."
The message quoted Ch Insp Roberts as saying the man had been taken, unconscious, from the ground and was in a critical condition at the University Hospital of North Staffordshire.
He urged anyone with any further information to contact Staffordshire Police.
A statement on Stoke City's website said police were investigating the incident "which has left a Blackburn Rovers supporter in a critical condition in hospital".
It said Ch Insp Roberts, who was the match commander, had confirmed the man suffered a "head trauma" in the incident which was confined to the visitors' section of the SEAT stand.
The statement said: "No further comment at this time will be made by the club as this is a matter being handled by the police."