The Cottagers started the 14th European match of their mammoth season knowing victory over Group K's bottom side would secure a place in the final 32.
But Fulham squandered victory three minutes into stoppage time as Fall headed home, resulting in Wisla Krakow finishing second instead.
The west Londoners' safe passage seemed all-but secured at the interval after two goals in four first-half minutes.
Excellent work down the left flank by Kerim Frei resulted in a low cross to Bobby Zamora, whose lay-off was finished by Clint Dempsey.
Four minutes later Mousa Dembele wriggled free and found Frei with a through ball, which the 18-year-old nonchalantly popped through Stefan Wessels' legs.
However, Fulham capitulated in the second period as Hans Henrik Andreasen's low free-kick and Fall's last-gasp header saw them drop out of the competition.
Heading into the match, Fulham were rocked by the news that goalkeeper Mark Schwarzer could miss up to six weeks after suffering a compression injury to his thoracic spine.
The injury saw 21-year-old Neil Etheridge make his first-team debut as one of seven changes Martin Jol made to the side that lost 2-0 at Swansea.
The match started edgily as Fulham's much-changed side looked to gel and their Danish counterparts struggled to maintain possession.
The first chance of the night came in the 10th minute when Chris Baird forced former Everton goalkeeper Wessels into action with a 22-yard drive.
Fulham continued to press after the midfielder's effort and, after fluffing a few chances, came close to breaking the deadlock through Brede Hangeland.
The Norway international connected with Frei's whipped corner and directed a header goalwards, but Bashkim Kadrii cleared off the line and Dempsey blazed over the rebound.
Zamora - whose relationship with manager Jol dominated the pre-match build-up - looked increasingly dangerous as the first period wore on.
Having been thwarted in a one-on-one with Wessels, Dempsey latched onto the striker's flicked header but saw his goal-bound effort cleared off the line by Espen Ruud.
Dempsey was not to be denied for long, though, as the Texan midfielder broke the deadlock in the 27th minute.
Frei burst down the left flank and fired in a low cross that Zamora laid off for Dempsey to fire home.
Fulham doubled their advantage four minutes later when Dembele played in Frei, who slotted through Wessels' legs from close range.
Kelly came close with a shot from an acute angle as the half-time break approached, before Zamora fired into the side-netting.
Dembele twice came close within three minutes of the restart as Fulham looked to begin the second period in the same vein, although Odense were beginning to get a foothold in the match.
Etheridge was forced into a fine save to thwart a low Ruud shot, before the Danish side almost capitalised on a melee in the box.
Odense's increased pressure saw them pull one back in the 64th minute when Andreasen found a gap in the wall and fired home a 20-yard free-kick.
Kadrii came close to drawing level moments later, leading Jol to bring on Damien Duff for Dempsey in a bid to alter the match.
The change did not have the desired effect, though, as Fulham continued to look lethargic and struggled to quell the waves of Odense attacks.
Jol's side looked uncharacteristically shaky in defence and lacked creativity in attack, leading to a nervy final 10 minutes.
Aaron Hughes headed over and Marcel Gecov shot wide as Fulham looked to put the game out of sight, before substitute Fall headed home three minutes into stoppage time at the end of a sweeping move to put Fulham out of Europe.