Filham, who had led Group E before the match, went ahead thanks to a 24th-minute header from defender Brede Hangeland, who outjumped Marco Andreolli to nod home a corner by Bjorn Helge Riise.
Midfielder Riise lined up against his brother, Roma defender John Arne, as the visitors struggled to make any impact without injured captain Francesco Totti.
Daniel de Rossi had an effort ruled out for offside in the 53rd minute, then Fulham had to battle after defender Stephen Kelly was sent off for a professional foul with 14 minutes after bringing down Riise -- but not before referee Paul Allaerts incorrectly handed a red card to Hangeland.
The furious Fulham players protested and Kelly was eventually dismissed, then goalkeeper Mark Schwarzer saved the resulting penalty from Jeremy Menez.
But it proved to be a brief respite as Andreolli leveled in the fourth minute of time added on with the last kick of the game as he stole in unmarked at the far post to strike from close range.
The result left Roma in third place on four points, while Swiss side Basel went top on six following a 2-0 victory away to CSKA Sofia.
Striker Alexander Frei netted in the 20th and 63rd minutes in Bulgaria to put his team a point above Fulham.
