Jason Scotland, the Trinidad and Tobago striker who had earned the Championship side a replay ten days earlier, opened the scoring two minutes after the interval.
And the noisy Welsh travelling support thought they had done enough to claim another Premier League scalp - they knocked Portsmouth out a round earlier.
But with 13 minutes left Davies, a Wales international, flung in a corner that Clint Dempsey nodded home. Three minutes later Davies assisted Zamora and, after putting up a great fight, Roberto Martinez's side were sunk.
Fulham manager Roy Hodgson had witnessed his side defeat West Brom 2-0 only 52 hours before kick-off and one could understand his complaints to the Football Association about his players being fatigued.
The replay had been moved forward a day by London Underground as they feared transport problems with local rivals Chelsea hosting Juventus in the Champions League on Wednesday.
Though he rested top scorer Andy Johnson and benched captain Danny Murphy, most of Hodgson's other first teamers started but took a while to shake off Sunday's bruises.
Scotland and Jordi Gomez came close but found Australian goalkeeper Mark Schwarzer in inspired form. Spanish loanee Gomez's low-struck free-kick on 28 minutes deflected off the bottom of the wall and, having already dived, Schwarzer did well to halt the shot with his trailing leg.
That close call seemed to shake Fulham from their torpor as they had the better of the remainder of the half. Dempsey crashed a free-kick just over before Erik Nevland's effort inside the area was well parried by Dorus de Vries.
Colossal central defender Brede Hangeland came closest to breaking the deadlock five minutes before the interval. But his thumping header from Davies' corner only found the Swansea crossbar.
However, it was the visitors who drew first blood through Scotland. The 30-year-old striker was found by Mark Gower just inside the Fulham box and, with too much space for Hodgson's liking, toe-poked his effort to the left of Schwarzer's groping hand - it was his 14th goal in 15 games.
Hodgson was forced to introduce skipper Murphy on the hour mark after Olivier Dacourt hobbled off. The former England midfielder could have levelled eight minutes later when Zamora's cut-back found him free only six yards out but the 31-year-old contrived to side-food his chance wastefully over.
The home side's persistence finally paid off when American Dempsey nodded in Davies' corner in the 77th minute.
Zamora competed the scoring soon after and earned his side their first FA Cup quarter-final in five years. They host Manchester United in the last-eight. Hodgson will be hoping his team, and goal machine Zamora, are well rested.