Extra-time goals from Collins John and Claus Jensen booked Fulham an FA Cup fifth round tie against Bolton.
Gregorz Rasiak gave Derby an early lead with a close-range strike in an opening half which was dominated by the Rams.
Fulham levelled when Papa Bouba Diop netted from the spot after Michael Johnson handled and Luis Boa Morte's right-foot shot made it 2-1.
Paul Peschisolido bundled in Derby's second but John's strike and Jensen's tidy finish won it in extra-time.
It was, perhaps, a cruel end for Derby, who were forced to play the last 22 minutes with 10 men having lost Johnson to injury after using all their substitutes.
Particularly as it was Fulham who had been caught short as the Rams made a flying start to the match.
Derby broke quickly and Rasiak got in at the near post to meet Tommy Smith's cross to give the visitors the lead.
However, the Championship side were not prepared to rest on their laurels and continued to press. Within the first 20 minutes Rasiak saw his header hit the post and Morten Bisgaard's long range strike was saved.
Such was Derby's dominance in the opening half-an-hour that Fulham failed to muster an effort on goal.
But the Rams' slender lead was cancelled out on the stroke of half-time when Fulham were awarded a penalty.
Johnson, while attempting to make a sliding tackle, handled and Diop coolly stroked the spot-kick into the top right-hand corner.
Fulham, boosted by their late leveller, began the second period a different team and were positive and were rewarded when Boa Morte gave them the advantage with a well taken right-footed shot.
Boa Morte, buoyed by his first goal, had the appetite for the fight and almost added a second with a powerful header which was well saved by Lee Camp.
Lee Clark, who was anonymous early on, was pulling the strings in midfield.
Derby kept going and continued to pose an attacking threat particularly when Bisgaard's shot was repelled by Van der Saar.
Their enterprise bore fruit when substitute Peschisolido capitalised on indecisiveness in Fulham's defence to poke in from inside the box to set up extra-time.
Another substitute John made his mark when he swivelled in the area and drilled his left-footed shot past Camp.
The Premiership side increased the tempo and substitute Jensen broke clear of a tiring defence to slot home.
Fulham: Van der Sar, Rosenior, Rehman (Goma 45), Knight, Bocanegra, Radzinski, Diop, Clark, Legwinski (Jensen 102), Boa Morte, Cole (John 80). Subs Not Used: McBride, Crossley.
Goals: Diop 45 pen, Boa Morte 50, John 94, Jensen 105.
Derby: Camp, Kenna, Mills, Johnson, Boertien, Tudgay, Bisgaard, Idiakez, Bolder (Peschisolido 83), Smith (Junior 28), Rasiak, Junior (Doyle 90). Subs Not Used: Grant, Turner.
Goals: Rasiak 4, Peschisolido 86.
Ref: P Walton (Northamptonshire).