Bognor are through to the next round, although for long periods of their game at Uxbridge it looked like they were going to need a replay.
They had to wait until the 85th minute to score the only goal, Matt Russell latching onto a mistake and gleefully volleying home.
Rocks controlled the opening 45 minutes but clear chances were at a premium although Russell had an effort blocked from close range.
Guy Rutherford injected some venom and his long range effort stung the hands of Ray Francis in the home goal, who was also able to deny Russell from the rebound.
Minutes later, substitute Rupert Sansom sent in a rasping drive that beat Francis but flew wide.
Uxbridge came close to stealing a win deep in the second period when Lee Tunnell was denied by Craig Stoner.
Bognor manager Jack Pearce suggested he would have settled for a replay. He said: "The first priority is to be sure your name is in the hat, anything after that is a bonus."
Worthing face a replay at Cambridge City tomorrow although they had the better of their goalless clash against the Dr Martens Premier Division side at Woodside Road.
Top scorer Sam Francis, who shot wide when clean through in only the third minute, also put a downward header over just after the interval from Mark Pulling's corner.
Rebels finished strongly and penalty appeals for handball were waved away following a shot by Paul Rogers who also had a late effort ruled out for a push on goalkeeper Alan Calton.
Worthing found 17-year-old Calton in fine form and he denied Gavin Geddes and Mark Knee in quick succession after the break. Calton also did well to thwart Knee in the first half when producing a superb one-handed save.
Eastbourne Borough boss Garry Wilson thought his side were hard done by after they went down 2-1 at home to Welling United.
He said: "There was no way we deserved to be 2-0 down at half-time, it was just a bit of naive defending.
"We played the better football and we completely dominated the second half. There is no way we deserved to lose."
Both Welling goals came against the run of play. Danny Slatter gave Welling the lead on 11 minutes when he slammed home from 12 yards after Borough had failed to clear a cross.
Then on 41 minutes, Paul Booth headed home unmarked from close range. Borough dominated the second half but their only reward was a header from close range by Danny Simmonds.
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