Callum Brittain missed two chances to give Barnsley the lead in the first half. The midfielder’s close-range effort from Alex Mowatt’s cross in the 10th minute forced a superb reaction save from Kepa Arrizabalaga. Then his shot following a clever corner routine just before halftime was deflected off target.
In between, Chelsea was denied a penalty by referee Martin Atkinson.
Abraham galloped onto Billy Gilmour’s through ball and Barnsley defender Toby Sibbick caught the Chelsea striker’s foot as he slid in for a goal-saving challenge.
Chelsea had not managed an effort on target by the hour mark but took the lead when James ran down the right flank and squared for Abraham to sidefoot into an empty net.
Replays showed Abraham may have been offside when James played the ball across, but there is no VAR used in the FA Cup.
Abraham then rescued Chelsea at the other end, bailing out Arrizabalaga.
The Blues goalkeeper failed to get anywhere near a free kick into the area but Abraham was on hand to head Barnsley substitute Michael Sollbauer’s hooked effort off the goal-line from under the crossbar.
American forward Daryl Dike came off the bench in the 59th minute to make his debut for Barnsley. The 20-year-old Dike is on loan from MLS club Orlando City.