Gomes made a string of fine saves as the Cottagers went on the rampage after the interval to book a replay at White Hart Lane on Wednesday March 24.
The Brazilian prevented Bobby Zamora, Damien Duff, Brede Hangeland and Zoltan Gera scoring in a London derby that lacked any bite until a pulsating second period.
For Gomes, it was a return to the ground where in November 2008 he made the worst in a string of blunders that almost saw him sold by manager Harry Redknapp.
Redknapp opted against offloading the £7.8million signing from PSV Eindhoven.
The Spurs manager's faith has already been repaid but once more Gomes underlined his value.
At the other end, Mark Schwarzer kept out Niko Kranjcar and Peter Crouch at the end of the first half, but was hardly involved again.
For all their possession and a passing game that was easy on the eye, Spurs rarely troubled Fulham's goal and wilted as an attacking force.
Fulham manager Roy Hodgson will be disappointed his side, who continue to punch above their weight in two cup competitions, did not finish off their capital rivals.
Hodgson and Schwarzer were presented with their manager and player of the month awards for February shortly before kick-off.
Spurs pressed hard in midfield to give their opponents little time on the ball but despite some slick passing, they failed to trouble Fulham's goal.
Anticipation gripped home fans as Zamora broke free down the left before expertly cutting back inside where he fired a shot straight into the arms of Gomes.
It was not until the 28th minute that either keeper was extended, Schwarzer palming away a well-struck shot from Niko Kranjcar.
The big Australian then had to dive sharply to his left to turn a fine goalbound header by Peter Crouch away from the target.
Damien Duff pulled the trigger from long range at the other end as Fulham's chances continued to be severely limited.
Zamora bulldozed his way through at the end of one well-worked move but Michael Dawson intervened to snuff out the danger.
The second half was not even a minute old when Gera drew a superb save from Gomes.
Simon Davies initiated the move with a marauding run down the right followed by a pinpoint cross to the far post where Zamora directed the ball to Gera whose header was well stopped by Gomes.
The Brazilian then had to react sharply to keep out Hangeland's header from a Davies corner.
Dawson smashed the ball into the stands to end a spell of Spurs' dominance but once again the visitors did not look like scoring for all their dominance.
The match was coming alive after a dull first half, partly thanks to the noisy Spurs fans.
Duff found the side netting after being set up by Davies and was then denied by another fine save from Gomes following excellent work by Zamora.
The impressive Kranjcar fizzed a lethal cross through the penalty area as Spurs launched a swashbuckling counter-attack.
Space was opening up for both sides in an increasingly frantic final 10 minutes that saw Zamora fail to connect with a fine chance.
Vedran Corluka almost turned an attempted clearance into his own net - he did not know where the ball was going and Spurs breathed a sigh of relief when the final whistle went.