Robins fell behind early in the second half but sent the game to extra time thanks to a superb strike from Scott Williams.
Substitute Jake White then drew Bracknell level again after Fairford had scored against the run of play.
Neither side could grab a late winner so it was down to a shootout, which saw both sides converted four out of five spot-kicks.
Unfortunately for Robins, their sixth effort was saved and Fairford progressed to the next round.
Wokingham & Emmbrook beat Flackwell Heath 2-1 in the same competition thanks to goals from Luke Lee and George Dance.
Lee gave Wokingham the lead in the 20th minute and Dance doubled the advantage early in the second period.
Flackwell hit back with a late goal from Louis Cordice but they were unable to grab an equaliser and send the game to extra time.
Sandhurst Town were unable to inflict just a second defeat of the season on Combined Counties Premier Division leaders Guernsey as they won 2-0 at Bottom Meadow.