Tammy Abraham's second-half tap-in was enough for the Blues to claim victory at Oakwell.
The Championship outfit's best two chances both came from Callum Brittain in the first-half, one which was denied by Kepa and another, after a well worked short corner, which ended up narrowly missing the far post.
Tammy Abraham also cleared Michael Sollbauer's shot off the line.
However, the German will be frustrated with the performance of his team as they struggled to dominate in Yorkshire.
Regardless, the Blues are now into the final eight where they will play Sheffield United on March 20 or 21.