Five people were arrested when fighting broke out after the Middlesbrough v Sheffield United game on Saturday.
Police have said they expect to make more arrests.
One police officer was taken to hospital.
The problem began before the final whistle when Blades fans thought they had scored an equaliser but it was disallowed.
Middlesbrough won the game 1-0.
Assistant Chief Constable Jason Harwin said: “We will absolutely not tolerate violence of this nature and we will work to identify the offenders and bring them to justice.
“The vast majority of fans accept that football is a family game, watched by children, parents and grandparents.
“It’s terrifying for children to witness such violence, or be innocently embroiled in it as their parents frantically try to keep them safe.
“Unfortunately a small minority of people behave in a manner which puts innocent fans at risk.”