The Robins saw off Aylesbury-based Grendon Rangers 1-0 in their fifth round tie, but it was thanks to the committee members and supporters that the game even went ahead.
The pitch was still extremely muddy and not conducive to playing good football, which certainly proved a leveller for the match.
The Robins hit the woodwork on three occasions but Taylor Elliott-Money got the only goal when he headed in Jake White's cross in the first half.
Grendon had their chances and could easily have levelled, but Bracknell remained resilient to secure the victory.
A last-minute decision was made for Ascot United's match against Reading Town to go ahead and the Yellamen went on to win 4-1 at Scours Lane.
The only Hellenic Premier Division fixture to be played was made possible after Ascot's FA Vase game at Borrowash was postponed for the second time.
Both teams found it hard to play any constructive football in the first half as the pitch was heavy and bumpy.
In the second period Andy Lunn replaced Ian Brooks to join Ben Knight up front, and within a couple of minutes the visitors went ahead.
Dave Hancock surged into the box and rounded a defender, and his cross to the far post was headed in by Knight.
On 54 minutes it was 2-0 when Rob Saunders pounced on a loose ball and clipped the ball over the advancing goalkeeper.
Jon Bennett then replaced Knight and got on the scoresheet after 66 minutes when his low drive went through a crowd of players and beat the keeper.
Substitute Josh Warren rounded off the scoring with a fine solo effort.
James Knight, who recently returned to Reading, netted a late consolation for the home side.
Elsewhere, Binfield's match away to Highworth Town was called off but the Moles are due to play Thame United tonight (Monday, 7.45pm ko).
The Larges Lane pitch was too waterlogged for Wokingham & Emmbrook's match against Ardley to go ahead while Chalfont Wasps versus Finchampstead in Division One East was also called off.
In the Combined Counties Premier Division, Sandhurst Town were beaten 1-0 at Hanworth Villa.
Meanwhile, Eversley & California's Combined Counties Division One game against Cobham was postponed.
* For reports, pictures and previews to next weekend, pick up a copy of Thursday's News.