Having already recorded a 2-1 victory at Saltergate, a crowd of 8,417 saw the Reds complete a 6-2 aggregate victory over the Spireites.
Goals from Andy Liddell, Neil Redfearn, Darren Sheridan and Georgi Hristov sent Barnsley into the next round.
Two delightful chipped efforts came from Sheridan and Hristov in the second half.
Barnsley Player subscribers can watch the highlights here or Darren Sheridan's goal below on YouTube.