Winger Sam Barratt scored four goals for the Robins in a thrilling 6-4 defeat at a freezing Larges Lane.
There was also a debut for 47-year-old Mick Havermans, who played for the Robins in the nineties and whose son Joel plays for Binfield.
Bracknell went 2-0 up through a penalty and free-kick from Barratt, before Headington fought back to 3-2.
Barratt equalised from a free-kick again and then put the Robins 4-3 ahead after great pressure by Havermans.
With 20 minutes remaining, Headington hit back again with three goals to seal the tie.
Elsewhere, Sandhurst Town's match away at Bedfont Sports in Combined Counties Premier Division was postponed due to a frozen pitch.