In a hard-fought and even encounter at Loftus Road, Alan Pardew's men gave another splendid account of themselves away from home and deserved at least the replay that will take place at Upton Park on Tuesday, February 24.
The result also means that Hammers' name will be in the hat for the quarter-final draw on Monday, to keep confidence bubbling ahead of the vital trip to league leaders Norwich City back in the Nationwide next Saturday, when several players will return to the fold after sitting out today's visit to west London.
With no fewer than six players ineligible, the Hammers boss had been forced to reshuffle his side following last week's victory at Bradford City. His options were boosted by the return from injury of influential winger Matty Etherington, while Kevin Horlock and Anton Ferdinand came into the starting line-up to ensure only the slightest disruption to Hammers' current momentum.
With Ferdinand slotting in at right-back, the versatile Hayden Mullins was asked to adopt an unfamiliar left-back role, while Horlock joined Michael Carrick in the centre of midfield.
Despite the changes, Hammers made a blistering start and almost opened the scoring after just 18 SECONDS. Etherington immediately proved his fitness with a barnstorming run and deep cross that found Harewood unmarked at the far post, but the first division's leading scorer saw his fierce volley superbly kept out by Edwin Van der Sar.
If anything, that early scare seemed to provide a wake-up call for Chris Coleman's men, who - led by the dangerous Steed Malbranque - began to make their own forays into the Hammers' half.
Defender Volz saw an effort trickle just wide, Bywater had to be alert to keep out a rasping drive from Malbranque, while Bocanegra's looping shot bounced back off the post and was cleared to safety.
However, Hammers continued to enjoy a fair amount of possession and looked dangerous on the break, and Van der Sar was called into action again seven minutes before the interval, diving full length to keep out Deane's towering header.
The visitors began the second half in much the same way they had started the first, creating a glorious opportunity to open the scoring that once again had the travelling fans holding their heads in disbelief.
A looping cross from Etherington was only palmed out by Van der Sar, but Deane somehow managed to scoop the ball over the crossbar from just a few yards out.
Spurred on by the 3,000 magnificent supporters behind the goal they were attacking, though, Hammers continued to attack their Premiership opponents and, with Michael Carrick absolutely superb in the centre of midfield again, a shock was certainly on the cards.
Marlon Harewood saw a goalbound shot blocked by a defender, Christian Dailly headed inches wide from a corner, while the best Fulham could muster was a Zat Knight header that bounced well wide of the target.
The breakthrough finally looked to have arrived for Hammers with just seven minutes remaining. Another inviting cross from Etherington was only parried out by Van der Sar towards the penalty spot, where Carrick looked set to crown his performance with the winner but agonisingly saw the Fulham keeper block his effort.
The hosts rallied towards the end and Bywater was called upon again to tip a Legwinski effort round the post but, after substitute Steve Lomas had made a welcome return in his first appearance of the season, Hammers might have won it in the closing stages.
After Marlon Harewood's surging run had been crudely halted by a defender, the ball ran on to release Etherington clear on goal, but referee Graham Poll failed to spot the advantage and brought the play back for a free-kick - much to the annoyance of everyone in a claret and blue shirt.
That didn't take away from another stirring Hammers' performance, though, and Pardew's men will return to Upton Park for the replay confident they can get the job done at the second time of asking.