The Dutch keeper had to be at his very best to deny Marlon Harewood, Brian Deane and Michael Carrick and ensure his side remained in the hat for the quarter-final draw.
Despite being without six players, Alan Pardew's side showed they can still compete with the very best and will fancy their chances in the fifth round replay back at Upton Park on February 24.
Nigel Reo-Coker, Jobi McAnuff, Adam Nowland, Andy Melville, Bobby Zamora were all cup-tied while Jon Harley, on a month's loan from Fulham was not allowed to play.
West Ham should have opened the scoring within the first 30 seconds of the game.
Matthew Etherington raced down the left before crossing the ball into Harewood who, on the volley, forced a superb save from van der Sar diving low down to his left.
Fulham midfielder Steed Malbranque created his side's first opening of the game after five minutes when he made space for himself and fired a 25-yard drive which produced a comfortable save from Stephen Bywater.
The home side came close again on 13 minutes when Luis Boa Morte released Malbranque into the box and the Frenchman once again produced a fine save from Bywater, who dived away to his left to deny the Cottagers midfielder.
United forced their way back into a lively cup encounter and came close through Carrick in the 18th minute.
David Connolly released Carrick on the edge of the box and the England midfielder fired a fierce 30-yard shot which flew just inches wide of the post.
And the away side came even closer to breaking the deadlock after 22 minutes when Carrick turned provider for Kevin Horlock, but the former Manchester City midfielder saw his close-range shot well saved by van der Sar.
But it was Fulham who came within a whisker of opening the scoring on 27 minutes when Carlos Bocanegra's cross-shot beat Bywater in the West Ham goal, but the ball cannoned back off the post.
West Ham striker Deane forced another one-handed save from van der Sar on 37 minutes when Horlock's cross picked out the experienced forward inside the box, but his close-range header was superbly pushed away by the Dutch keeper.
Alan Pardew's side came out in the second half determined to get their noses in front and almost got off to a dream start on 47 minutes.
Etherington raced to the byline before planting a deep cross into the box, which van der Sar could only parry as far as Deane, but with the goal at his mercy he somehow managed to strike the ball over the crossbar.
Chances were few and far between in the second half, but both sides had late opportunities to win the game.
Bobby Petta's inswinging corner on 75 minutes fell perfectly for Zat Knight whose towering header flew just past the post.
And just two minutes later, Christian Dailly was desperately unlucky to see his close-range header for West Ham also fall narrowly wide of the goal.
But the best chance of the game came after 83 minutes when Etherington again caused all kinds of problems for the home defence down the left.
His cross into the box was palmed away by van der Sar straight into the path of Carrick. But the home keeper superbly pushed the Hammers midfielder's close-range shot to safety.
And West Ham were left infuriated just seconds before the final whistle when Etherington was clean through on goal, but referee Graham Poll refused to play an advantage and prevented the former Tottenham winger from snatching a possible late winner.
Man of the Match: Edwin van der Sar - The Dutch keeper produced a string of world-class saves to keep the Cottagers in the FA Cup.