Two goals in less than a minute from Connor Wickham and Ronan Murray just ten minutes into the game sealed the win for Town, against last years winners at Barnet's Underhill Stadium.
Town will now face Fulham on home territory in the next round of the competition with the date of the tie yet to be confirmed.
Town Side: Ranieri, Bennett, Dunbar, Meekings, Eastman, Hyam, Clark (Griffths 33), Cawley, Caron, Murray (Butcher 73), Wickham.