Martin Jol's side looked like they were on the way through, but a late collapse - combined with Wisla Krakow's win over FC Twente - sent them out
Goals from Clint Dempsey and Kerim Frei looked to have sealed the win for Fulham after a dominant opening 45 minutes, but a surge in the second half from the visitors with goals from Hans Henrik Andreasen and Djiby Fall sealed a last-gasp point for the Danes.
On a chilly night in on the banks of the River Thames, Fulham were playing some nice football in the opening stages. Frei in particular was keeping the ball well and evading opponents with ease.
The hosts went for a blend of youth and experience with injuries to key players Mark Schwarzer and Danny Murphy meaning starts for Neil Etheridge in goal and Marcel Gecov in midfield.
After some decent chances including a goal-line clearance from Kadril after Brede Hangeland's header was saved by Wessels.
The hosts eventually took the lead in the 27th minute after some fantastic wing play from Frei. The 18-year-old marauded down the left flank before finding Bobby Zamora in the box. His first time lay-off to Dempsey was dispatched with aplomb into the bottom corner.
And they increased their lead less than five minutes later as the impressive Frei was fed into the box by Moussa Dembele, and the small winger coolly placed his finish under the legs of Wessels.
It looked as if the match would be plain sailing for Fulham from this point, but Odense came out after the half-time break with renewed vigour, and the hosts were dropping deeper and deeper as they game progressed.
With 20 minutes gone in the second half, the away side nicked a goal back after Chris Baird made a foul near the Fulham area.
A slight bit of sportsmanship from Andreasen in moving the ball slightly gave him a better angle, and he used it to devastating effect as he curled an effort around the Fulham wall and into the bottom corner of the goal.
The Fulham fans were already on their team's back, sensing a change in the mood of their team. And it didn't help either as they conceded possession regularly.
By the end of the match they had dropped to inside their own penalty area, desperately defending their lead as Wisla Krakow also held a lead over group leaders FC Twente.
And with literally the last action of the match, the Danes found an crushing equaliser. A combination of Orlando Sa, on by this point as a late substitute for Zamora, and Frei lost the ball on the edge of the Odense area.
A quick break from the visitors resulted in a fine pass out to the wing from Peter Utaka to Espen Ruud. Ruud's cross was inch perfect onto the head of substitute Djiby Fall, and with six seconds of the game left, the striker nodded the ball into the corner of the net, and Fulham were out of the Europa League.
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