Manchester United target Saha proved his value to Fulham with a goal eight minutes after the visitors had gone ahead in the first half.
And then the unstoppable Frenchman grabbed an injury-time winner to leave Cheltenham devastated.
The Robins took a shock lead after just five minutes through a fantastic strike by midfielder Grant McCann.
The 23-year-old took his time before hitting a splendid 25-yard strike past Edwin Van der Sar.
The lead was held for just eight minutes however as Fulham levelled through top scorer Saha.
The Frenchman grabbed his 13th of the season as he latched on to a long floated ball from Sylvain Legwinski to lift the ball over visiting keeper Shane Higgs and into the back of the net.
Cheltenham were full of fight and almost retook the lead within minutes. Damain Spencer tried to capitalise on a mistake by Alain Goma, only to be denied by Van der Sar.
Saha twice went close as Fulham attempted to turn the game around, but it was Cheltenham who came closest to doubling their tally as Michael Duff's impressive header was tipped away by Van der Sar.
Bob Taylor headed over two minutes before the break, as the half ended with the Premiership side being booed off the pitch by their disappointed supporters.
Saha hit a speculative long-ranger into the arms of the Cheltenham keeper on 53 minutes and Zat Knight headed wide soon after as Fulham searched for a second goal.
Barry Hayles was brought on in place of Junichi Inamoto in a bid to turn up the pressure on the Division Three side and ten minutes later Fulham created their best chance.
Malbranque and Sean Davis combined well down the right side and the ball was fed to debutant Bobby Petta who fired across goal, but no Fulham player was able to get a finishing touch.
Moments later a John Finnigan effort whizzed past the post for Cheltenham and on 73 minutes sub Hayles hit the post with a header from an Andy Melville flick-on.
Cheltenham appealed for a penalty on 76 minutes when Knight denied Brian Wilson, who was bearing down on goal.
But Saha had the last laugh as he popped up in injury time to head a Petta corner past the despairing Cheltenham keeper to secure a fourth round place.