Three goals inside six minutes, sparked by Ethan Ennis and inspired by Jack Fletcher's double, turned the tie completely on its head.
The Tykes led through a first-half double from Vigal Yoganathan and looked set for the three points until Travis Binnion's side, spurred on by the typically noisy away supporters, hit top gear at Oakwell.
Yoganathan scored with 11 minutes on the clock, from Kryan Lofthouse's pass, and the pair combined again for the Welshman to beat Tom Heaton for a second time as half-time approached.
Ethan Wheatley and Ennis had decent attempts for the Reds but Heaton had to make one top save to thwart Josh Benson.