A spirited performance from Barnet was not enough as Slavi Spasov’s first half strike was the only difference between the sides.
Aymen Azaze kept the Bees in it with an excellent penalty save in the second half but despite plenty of chances, Dan Senda’s side couldn’t find that elusive equaliser against a strong home team.
Crooks’ early effort was well saved by the Azaze just six minutes in. Chances for Martyn Box and Rio Connell came and went before Spasov broke the deadlock with cool finish low into the bottom left corner of the Barnet net.
Okem Chime had his header tipped over before Connell beat his man only to fire straight at the Oxford keeper.
The home keeper was again called into action and pulled off a smart save when Chime latched on to a loose ball six yards out.
The Yellow’s were awarded a penalty straight after the break, the referee initially awarding a free kick after Dwight Pascal’s foul but later changing his mind.
Spasov stepped up but Azaze guessed right and got down well to keep it out.
The Bees ramped up the pressure but the Oxford defence stood firm, Adeniran had a sight at goal as time ticked away but his effort just went wide.
In stoppage time, Chime struck the crossbar before Wraynel Hercules was bundled over in the box but no penalty was given.
At the other end with the last chance of the game, Azaze had to stay alert and stood firm when Edwards only had the Belgian keeper to beat.
The Young Bees FA Youth Cup journey came to an end at the fourth hurdle.
Barnet: Azaze, Pascal (Douglas 89), Box (Francis 76), Tompkins, Pauw, Taylor, Chime, Adeniran (Murphy 89), Connell, Titchmarsh, Hercules.
Unused: Omore, Edwards