In one incident a group of United fans broke into a food kiosk at the Oakwell stadium trapping eight staff, while they stole cash from the till and took all the stock, a South Yorkshire police spokesman said.
"Police officers responded by arresting three people. Some of the fans who broke into the food kiosk also threw food items at the Barnsley stewards," he said, adding a police officer received a minor facial injury trying to arrest one person.
Two fans also invaded the pitch during the game, holding up the action during the fourth round tie for several minutes before stewards eventually removed them.
Holders Manchester United won the game 2-0 to reached the League Cup quarter-finals.
In a separate incident after the match, two people were arrested after jumping on to a railway line and stopping a Barnsley-Sheffield train, he said.
Police added that four home fans and four visiting supporters had been arrested after the various incidents.
In a statement on its website the FA said it was awaiting reports from a Crowd Control Advisor and referee Chris Foy.
"The FA will also be liaising with both clubs and South Yorkshire Police to gather evidence relating to events at Oakwell.
"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."
