The Cottagers knew three points in Holland would guarantee qualification from Group K, but they ultimately claimed none as Austrian substitute Marc Janko stabbed home from close range on 89 minutes.
Martin Jol's side struggled against the club he played for between 1979 and 1982, failing to muster a single shot in a first half that Twente dominated.
Fulham's poor first period was compounded by an injury to Damien Duff, although the west Londoners looked a more cohesive attacking unit after the restart.
Moussa Dembele limped off as Fulham pushed forward towards the end and Janko scored a minute from time.
The win was Twente's first against English opposition and things got worse for the Cottagers when Johnson was sent off in stoppage time, meaning he will miss the all-important final group match with Odense, which Fulham need to win to be sure of qualification for the last 32.
Jol named a strong Fulham line-up for the game as they looked to qualify with a game to spare.
Duff, Johnson, Aaron Hughes and Stephen Kelly were all drafted in, with Bryan Ruiz among those dropping out.
The Costa Rica international - a former Twente favourite - was not involved in the match due to ineligibility but was honoured before the game kicked off by the Tukkers.
Nacer Chadli had the first shot of the match after 13 minutes with a drive that Mark Schwarzer saved down to his right.
The reported Fulham target tested the Australia international again moments later, before Leroy Fer unleashed a fierce shot from the edge of the box that Schwarzer did well to stop.
While the west Londoners were looking organised at the back, they were struggling to find a rhythm and get the ball out of their half.
Fulham were forced to alter their flat-looking attack in the 34th minute when Duff was forced off with a knock.
Kerim Frei replaced the Republic of Ireland international and the visitors soon eked out their first chance, with Johnson thwarted by the onrushing Nikolay Mihaylov when through on goal.
The visitors started the second period brightly and six minutes after the restart Kelly's cross was headed wide by Johnson under pressure from Peter Wisgerhof.
Twente were also attacking with intent and brought on Ola John - brother of former Fulham striker Collins John - to help utilise the space down the flanks.
The 19-year-old was proving dangerous down the left as Twente began to assert their authority once more, with Emir Bajrami shooting just wide from 25 yards.
Chadli and Douglas both attempted long-range efforts with the latter coming close before Johnson wriggled free in the box at the other end only to snatch at the ball and shoot across the face of goal.
Play was starting to sway from end to end and, after a Frei cross caused trouble in the Twente box, Luuk de Jong wasted a gilt-edged chance to break the deadlock as he headed wide from six yards.
Both sides were looking dangerous going forward and Bobby Zamora was unfortunate to direct a well-worked Danny Murphy free-kick wide in the 71st minute.
Fulham pushed forward in numbers as rain began to pour down in a final 10 minutes that saw Dembele limp off.
Things got worse for the visitors when neat play from Twente saw Janko turn home Roberto Rosales' cross.
The late goal was compounded further by Johnson's sending-off in stoppage time for a second yellow card after he kicked the ball away.