The hosts looked very flat in the opening period, and offered very little as an attacking threat, with Fulham very much the better side.
Jermain Defoe gave the home side more impetus after the break and posed a real threat.
However, Sunderland’s task was made more difficult after Jack Rodwell was given his marching orders after picking up two bookings.
Fulham couldn’t capitalise on their numerical advantage, but will be pleased to still be in the hat for the fifth round draw,
The Black Cats started brightly with wing-backs Billy Jones and Van Aanholt getting forward well, but they soon wilted as Fulham worked their way into the game.
In truth, neither side was able to create much of note with Marcus Bettinelli kicking away Sebastian Larsson’s 33rd-minute drive and Steven Fletcher looping a tame header over the bar five minutes before the break.
Home keeper Vito Mannone was equally under-employed, beating away Moussa Dembele’s thumping 22nd-minute effort either side of fielding tame attempts from Cauley Woodrow and Konstantinos Stafylidis.
Indeed, the most notable incident of the first half was a Rodwell tackle on Stafylidis with several studs on show which resulted in a booking.
The home side returned in determined mood and pinned the visitors back in an early flurry, although the initial impetus soon abated and it was Fulham who went close six minutes into the half when midfielder Seko Fofana wrong-footed defender Santiago Vergini and fired across the face of goal and just wide of the far post.
With the home fans among a crowd of just 22,961 starting to voice their frustration and the Londoners sensing an opportunity, the game opened up and Defoe was presented with his first half-chance, a long-range effort which he screwed well wide of the target.
The Black Cats forced a series of corners as they started to gain the upper hand, but they were unable to make the most of the pressure as Fulham defended manfully.
However, Bettinelli had to bundle Defoe’s 62nd-minute volley around the post after he connected with Jones’ right-wing cross six yards out, and the keeper came to the rescue once again after the striker got a toe to Rodwell’s scuffed shot three minutes later.
That proved to be Rodwell’s last contribution to the game when he was booked for a second time for preventing the keeper from throwing the ball out seconds later.
Bettinelli denied Defoe once again from distance with 18 minutes remaining and Van Aanholt drive a rising effort just over the bar six minutes later as the home side made their final push.
Connor Wickham tested Bettinelli with a fiercely-struck 82nd-minute effort, although it was fellow substitute Hugo Rodallega who closest to a winner when he rattled the post from 25 yards with four minutes remaining.