Eight catering staff were trapped inside a food kiosk and had to barricade themselves in a store room to escape the violence as fans broke in, stealing money and food and causing thousands of pounds worth of damage.
Rowing insists Barnsley now have no alternative other than to install grills in all visiting bar areas to protect their staff following the incident that marred United's 2-0 win.
He said: "Last night I was disappointed, today I'm very angry.
"We have pre-match briefings with the police, stewards and clubs before every match and look closely at every risk assessment.
"But last night's incident came out of the blue. How do you legislate for the behaviour of idiots?
"What concerns me is not so much the damage, but the way these thugs treated our young staff.
"Manchester United in general have very well-behaved fans and set good examples, but last night there was a minority group hell-bent on causing trouble because they could not have what they wanted.
"We did not allow the sale of alcohol at half-time and they couldn't accept our decision.
"The real Man Utd fans have apologised to our staff and it happens so rarely, but we now have to address the situation.
"The machinery can be replaced, but with staff it's more difficult to do that. We need to offer them more protection and will be putting grills up.
"It's a shame in our society that we are controlled by small groups. It's mob rule and we have no option other than to protect our staff in future behind iron bars.
"The cost of the damage will run into thousands of pounds, but it's not so much that as the effect it has had on our catering staff."
A South Yorkshire Police statement said: "Police responded quickly and effectively to a spontaneous disorder caused by a small group of people that had gathered near to the central bar on the North Stand.
"Officers in protective uniform were deployed to restore order, disperse the crowd and ensure the safety of staff inside the area."
Two men aged 22 and 24 were arrested for public order offences, a 26 year-old-man was arrested for theft and a 32-year-old man was arrested for throwing a missile onto a football pitch. All four men are from Manchester.
In a separate incident, three Barnsley fans were arrested for running on to the pitch during play.
An FA spokesperson said: "There is no place for anti-social or violent behaviour in our game.
"Any culprit found to be involved in these disturbances and identified as such by either the police or clubs should be subject to a lengthy ban from attending football matches."
A Barnsley spokesman added: "Scene of crime officers are down here now going through the North Stand and we're proceeding with our own investigation as well.
"It's a hell of a mess down there. Doors have been kicked in, tills have been damaged and there's food and drink everywhere. It looks like a swamp in one of the bars."