Travis Binnion's side were excellent in the first half and unlucky to be pegged back before the break when David McGoldrick converted a penalty to cancel out Jack Fletcher's seventh-minute opener.
The League One side upped their game following the restart, with McGoldrick completing a hat-trick, either side of a Chido Obi strike that was well received by the 382 Reds fans in the away section of Oakwell.
Jack Fletcher's red card further hampered United's cause and the South Yorkshire side took full advantage, with Neil Farrugia and Jon Russell generating a 5-2 scoreline that would not have been believable judging by the opening period.