The League Two side equalised within four minutes as top scorer Michael Symes tucked away his 14th goal of the season, but they suffered a setback in the 41st minute when Darran Kempson was sent off after bringing down Zoltan Gera.
Second-half strikes from Damien Duff and Gera sealed victory for Fulham as the Premier League team made the most of their man advantage.
Roy Hodgson named a strong side for the tie, making only three changes from the team that lost 2-0 at Blackburn last weekend.
Andy Johnson was missing with a knee injury so Nevland started up front, while Bjorn Helge Riise and Toni Kallio were also brought into the first XI.
Accrington boss John Coleman kept faith in the same team that had beaten Gillingham on Tuesday in the third round.
Stanley, the lowest-ranked side remaining in the competition, knew a giant-killing would earn them a place in the fifth round for the first time in their history.
And with the club having experienced severe financial difficulties in recent times, there was an extra incentive for Coleman's players to try and upset the odds against their opponents and secure a lucrative fixture either back at Craven Cottage or in the next round.
Accrington started well, and Symes rifled in an effort in the ninth minute from 25 yards which Mark Schwarzer did well to tip around the post.
Five minutes later James Ryan met a bouncing ball with a volley but did not have the power to cause Schwarzer problems and the Australian made a comfortable save.
Stanley appeared to be making a mockery of the 70 places between themselves and Fulham, but after all the hosts' endeavour, it was the Cottagers who then took the lead in the 21st minute.
Duff sent in a corner which Gera flicked on and Nevland finished from point-blank range.
Accrington were back in it within four minutes as Symes pounced after a goal-mouth scramble.
Schwarzer could only parry Ryan's strike and Sean McConville then had a shot blocked by Brede Hangeland, but Symes was on hand to finish and draw things level once more.
It was nothing more than the Reds deserved, and roared on by a vociferous home support they went in search of a second goal with Robert Grant bringing another save out of Schwarzer shortly after the half-hour mark.
Trouble struck for Accrington just before the break though as they were suddenly reduced to 10 men, defender Kempson the man dismissed after going in from behind on Gera.
Having made it to half-time, Stanley emerged for the second half looking determined Kempson's red card was not going to ruin their day.
The hosts did their best to maintain the tempo and almost took the lead in the 52nd minute when Grant connected first-time with Ryan's cross, only to see his volley flash across Schwarzer and wide.
But the Cottagers were settling into a rhythm, and after he just failed to find a finishing touch to a neat passing move through the Accrington defence, Duff popped up again in the 58th minute to put Fulham 2-1 in front.
The goal was a little fortunate, the Irishman sending a weak effort skidding across the mud that Dean Bouzanis got a hand to but could not prevent from bobbling over the line.
Nevland then unleashed a fierce shot at Bouzanis from inside the box which was tipped over the bar, and Aaron Hughes could not get his header on target from the corner that followed.
Stanley were looking increasingly weary and it came as no surprise when Fulham extended their lead with 11 minutes to go.
Riise broke clear on the right and provided the cross for Gera, who had no trouble meeting it first time and sending past a helpless Bouzanis.
Source: Team Talk