The brave Third Division outfit more than held their own after taking an early lead through Grant McCann, and were well worthy of a second bite of the cherry, but were ultimately undone by Saha's predatory instincts.
The visitors made an explosive start, stunning the home crowd, when former West Ham midfielder McCann lashed home a memorable opener.
The Northern Ireland international dispossessed Sylvain Legwinski twenty-five yards from goal and powered in a left-footed drive which deceived Edwin van der Sar in the air.
Fulham almost found an instant reply, but top-scorer Saha could not generate enough power with a header from Junichi Inamoto's flighted cross.
However, the Frenchman did bag his 13th goal of the campaign on twelve minutes when he latched on to Legwinski's speculative long ball before coolly lobbing the stranded Shane Higgs from fifteen yards.
The Robins refused to be bowed by the leveller, though, and continued to give their more illustrious hosts as good as they got.
Indeed, just past the half hour mark defender Michael Duff forced van der Sar into an excellent stop with a firm header from a McCann free-kick.
The visitors passed up another good headed opportunity shortly before the break when veteran striker Bob Taylor nodded over from six yards after a left-wing corner had been knocked back across the box.
Fulham did begin to get a grip on proceedings after the interval and almost moved ahead on the hour mark.
After a neat interchange with Steed Malbranque, Sean Davis released debutant Bobby Petta on the left hand edge of the box but the Dutchman's drilled cross narrowly eluded the onrushing Saha.
The hosts then rattled the woodwork on 73 minutes when substitute Barry Hayles redirected Andy Melville's header against a post from close range, though Cheltenham saw claims for a penalty waved away when Zat Knight recovered to outmuscle Graham Fyfe in the box minutes later.
However, just as the Town looked set to secure the reply their brave efforts merited, Saha popped up in the dying moments to angle in a precise bouncing header from Petta's left-flank corner.
Taylor forced van der Sar into a sharp save with the last kick of the game, but John Ward's charges will feel shattered not to be in the hat for the fourth round draw after a heroic performance.
