Roy Hodgson's side - who came seconds away from defeating the Romans at Craven Cottage a fortnight ago, took a surprise lead in the Italian capital after just 19 minutes, with Diomansy Kamara scoring from the spot after he had been upended by Marco Andreolli.
However, the tie turned in the hosts' favour after Erik Nevland was shown a straight red-card in the 49th minute for a foul on Danielle de Rossi.
And after John Arne Riise and Stefano Okaka Chuka had completed the turnaround, Paul Konchesky also saw red late on.
The defeat sees Fulham drop to third place in Group E with two games remaining. The Londoners trail Roma by two points and group leaders Basel by four.
"With these types of refereeing decisions it's very hard to get a result," Fulham manager Roy Hodgson told ITV.
"We hope the referees will be a bit kinder to us in the Premier League than in this league."
On Merseyside, Portuguese heavyweights Benfica followed up a 5-0 rout of Everton in Lisbon with another victory over David Moyes' men, this time triumphing 2-0 at Goodison Park courtesy of goals from Javier Saviola and Oscar Cardozo.
The visitors also hit the post through Cardozo, while goalkeeper Tim Howard brilliantly denied Saviola in a convincing win for Jorge Jesus' side.
However, the Toffees still lie second in Group I with six points, two ahead of BATE Borisov and AEK Athens.
Elsewhere, Celtic are still looking for their first Europa League victory after holding Hamburg to a goalless draw in Germany.
The Bhoys must now win their remaining two games and hope Hamburg lose both of theirs if they are to qualify for the knockout stages.
Last season's UEFA Cup winners Shakhtar Donetsk won 2-0 at Toulouse to make it four wins out of four for the Ukrainians.
While Spanish side Villarreal, having endured a torrid start to their la Liga campaign, followed up their 5-0 weekend win over Tenerife with an emphatic 4-1 victory over Lazio at El Madrigal.
NEWS: Shakhtar march on
Results in full:
AEK Athens 2-2 BATE Borisov
Anderlecht 3-1 Timisoara
Basel 3-1 CSKA Sofia
CFR Cluj 2-3 Sparta Prague
Dinamo Bucharest 0-3 Galatasaray
Dinamo Zagreb 0-2 Ajax
Everton 0-2 Benfica
Copenhagen 1-1 PSV
Twente 2-1 Tiraspol
Fenerbahce 2-1 Steaua Bucharest
Genoa 3-2 Lille
Hamburg 0-0 Celtic
Heerenveen 2-3 Hertha Berlin
Levski Sofia 0-1 Salzburg
Nacional 1-1 Athletic Bilbao
Partizan Belgrade 2-4 Club Brugge
Rapid Vienna 0-3 Hapoel Tel-Aviv
Roma 2-1 Fulham
Sturm Graz 0-1 Panathinaikos
Slavia Prague 2-2 Valencia
Sporting 1-1 Ventspils
Toulouse 0-2 Shakhtar Donetsk
Villarreal 4-1 Lazio
Werder Bremen 2-0 Austria Vienna
