Five people have so far been arrested but police expect further arrests to be made as the investigation continues.
Cleveland police said the violence, which has also led to an officer being hospitalised, is “completely unacceptable”.
Away fans descended onto the Middlesborough stadium pitch just before the final whistle.
The invasion sparked the initial violence among fans from both sides which led to a riot in the coach park – dragging innocent children into the brawl.
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.
“We will be working closely with Middlesbrough Football Club and Sheffield United as part of our investigations.”
Those already in police custody have been arrested on suspicion of a number of offences.
Middlesborough won the Championship game 1-0.