The League One side never looked like claiming another Championship scalp to match that of Preston from the opening round of the competition.
Defender Keith Lowe celebrated extending his loan period at Turf Moor from Wolves by heading the Clarets in front as Garreth O'Connor's corner was flicked on by the head of Graham Branch for the 19-year old to convert at the far post.
Two goals in the space of seven minutes killed off the tie as striker Ade Akinbiyi claimed his seventh goal of the season with a delightful first touch in controlling a long clearance from Danny Coyne and rounding helpless Tykes keeper Scott Flinders.
Akinbiyi then turned provider to set up John Spicer for his first goal on his full home debut since his transfer from Bournemouth.