A host of changes from the Blues included amongst others goalkeeper Kepa Arrizabalaga, Emerson Palmieri, N'Golo Kante, Christian Pulisic and Abraham.
Chelsea were given a big scare in the opening ten minutes through a close-range shot from Callum Brittain after a good move into the area, but Arrizabalaga pulled off a fine stop.
However, having soaked up all the pressure the Blues began to push forward moments later and threatened the lead with a snap-shot from the far side of the area by Pulisic.
Barnsley should've gone to halftime leading from a short corner-kick on the right-side, but Victor Adeboyejo failed to turn a scuffed shot from Brittain into a gaping net as the first half concluded.
Two changes from the Blues ahead of the second half saw Andreas Christensen sacrificed for Antonio Rudiger and Reece James installed for Marco Alonso.
Barnsley lived dangerously following the opening minutes of the second stanza, as Callum Hudson-Odoi smashed a low shot from the shoulder of the area across the far corner of the net.
Rudiger could've announced his entry in style following 60 minutes with a pin-point header from a near post free-kick, but he nodded off target.
Chelsea were ahead five minutes later after Abraham strayed into the near corner to tap home a first-time cross from James, amidst offside protestations.
The Premier League giants nearly surrendered the lead 12 minutes from time following a tap from a corner by Michael Sollbauer, but Abraham produced a fantastic goal-line clearance.
Barnsley launched numerous threats as fulltime loomed, but the Blues held on to claim victory and a quarter-final date with Sheffield United.
Everton v Manchester City
Bournemouth v Southampton
Leicester City v Manchester United
Chelsea v Sheffield United