Children as young as 10 were injured after a brawl broke out between more than 200 football fans in a coach park.
Five people were arrested in the wake of the fight after the Middlesbrough v Sheffield United game on Saturday.
Cleveland Police said youngsters were “terrified” by the violence, which left one police officer in hospital.
Disorder began just before the final whistle when Blades fans thought their team scored a last minute equaliser, only for it to be disallowed.
Those held remain in custody while inquiries continue.
Cleveland Police Assistant Chief Constable Jason Harwin said the force would “absolutely not tolerate violence of this nature”.
He added: “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.”
Middlesbrough won the Championship game 1-0.