Police are investigating "co-ordinated disorder and criminal damage" before, during and after Bristol City's match against Swansea City on Saturday.
Officers used dogs and horses to separate the fans and said there was damage "to at least two pubs".
Ch Supt Paul Mogg said a large group of Swansea fans arrived "with the intention of drinking to excess and seeking confrontation".
He said three officers needed medical attention and a dog was injured.
Police and stewards were attacked and a number of people were reportedly bitten by police dogs.
In a statement, Bristol City FC said some 800 Swansea fans arrived 20 minutes after kick-off "and immediately removed the segregation nets on the east corner of the Atyeo Stand".
"Twenty-five Swansea City fans were removed during the game and there were a number of arrests.
"Regrettably, several Bristol City supporters were involved in altercations in the corner of the Atyeo Stand and Dolman Stand before being apprehended.
"The incident in its entirety was captured on CCTV and a formal investigation in conjunction with the Avon and Somerset Police match commander and our head of safety is under way."
Swansea City AFC is yet to comment.
Police and stewards were attacked and a number of people were reportedly bitten by police dogs
Ch Supt Mogg said there was disorder within the ground, with objects thrown by each group of supporters. He said there was further disorder after the match in the nearby Greville Smyth Park.
"We arrested a man from Swansea for assaulting another man from Wales," he said.
Police escorted Swansea fans from Temple Meads railway station to Ashton Gate stadium
He added that an officer was struck in the face by a coin, and a mounted officer and a colleague on foot were pushed into a fence.