Eight catering staff were trapped inside a food kiosk and had to barricade themselves in a store room for around 25 minutes as fans broke in and stole cash from the till and food from the shelves.
Stewards on the pitch also had food thrown at them during the violence, marring United's 2-0 fourth-round win.
The spokesman said: "No-one was injured during this time but one police officer suffered a minor facial injury.
"Eight fans were arrested, four Manchester United fans and four Barnsley fans.
"They have been taken to Barnsley police station."
A heavy on-pitch police presence was visible during the trouble.
Police in riot gear were pelted with bottles as they attempted to restore order in the concourse of the north stand and police dogs were employed to drive fans back from the pitch.
The police spokesman also said that around 11pm two more people were arrested after fans jumped on to railway tracks in a bid to stop the Barnsley to Sheffield train before it had started moving.
Barnsley said on Tuesday night that they would launch an investigation into the violence, accusing Manchester United fans of causing "substantial damage" at the stadium.
The club has complained to police, who are to carry out their own inquiry into the incident.
A Barnsley spokesman said: "The club is going to launch a full investigation as soon as possible after substantial damage was caused in the concourse of the north stand.
"A complaint has been made to the police and we will be taking a look at CCTV footage of the incident before we can comment further."