A late goal from FC Twente substitute Marc Janko means 10-man Fulham must win their final game to guarantee a spot in the last 32 of the Europa League.
The Cottagers were hanging on for a draw until Janko converted Roberto Rosales's cross two minutes from time.
Martin Jol's men will rue near misses for Andrew Johnson and Bobby Zamora - and Johnson was then sent off for a second yellow card in stoppage time.
He will now miss the final game, which is at home to Odense on 14 December.
Twente have lost just one match in the Dutch league so far this season, and it was easy to see why they are so hard to beat as they controlled possession well, while Bobby Zamora found it hard to break free from the shackles of Douglas in the early stages.
The home side lined up with a three-pronged attack and it was the left of the trio - Nacer Chadli - who caused the Cottagers problems, twice forcing Mark Schwarzer into saves before curling in a ball which Luuk de Jong laid off for Leroy Fer, but Schwarzer was again equal to the task.
Fulham had made four changes from their weekend draw at Arsenal, but with Liverpool next on their agenda in the Premier League, they will have been concerned to see Damien Duff last just over half an hour before succumbing again to the calf injury which forced him out at the weekend - and Jol confirmed he is likely to be without his Irish winger against the Reds after the game.
For all Twente's class on the ball, chances were at a premium and at the other end, Fulham offered next to nothing in a first half which will not live long in the memory.
A familiar name for Fulham fans arrived at the half-time interval in the shape of Ola John, the younger brother of their one-time striker Collins - but the visitors showed more inventiveness in the early exchanges of the second half.
It had been more than five hours since the Cottagers had scored a goal for themselves, but Johnson went closest with a header from Stephen Kelly's cross.
John almost created a chance for the home side but just as Rosales steadied himself to volley his cross goalwards, De Jong nicked the ball off his team-mate's toes.
Emir Bajrami whistled a shot just past the upright, before Douglas tried to take matters into his own hands, running through from halfway, but his long-range shot was off target.
The stalemate could then have been broken, with gilt-edged chances spurned at both ends, as first Johnson blazed wide after being inadvertently played in by a ricochet off Douglas before John's cross saw De Jong beat the offside trap but somehow he headed wide from six yards.
Zamora prodded Danny Murphy's clever free-kick wide, but the breakthrough finally came when Rosales got free on the right and Janko could not miss from four yards for his 17th goal of the season.
Fulham's misery was compounded in stoppage time when, having been booked earlier for a foul on John, Johnson was sent off for kicking the ball away, meaning he will miss the crucial final group game against the Danes.