Denes Rosa ended Hearts' participation in the Uefa Cup by grabbing the winning goal for Ferencvaros at Murrayfield.
Hearts needed a victory to have a chance of qualifying from Group A and Paul Hartley had the ball in the net only to be pulled up for hand ball.
Ferencvaros goalkeeper Lajos Szucs also produced a couple of good saves to keep the Scots at bay.
And Rosa stroked in the winner at the third attempt after two blocks by goalkeeper Craig Gordon on 30 minutes.
Hearts, the last of Scotland's participants in European competition this season, had gone into the game hoping they could win while Basle failed to defeat Feyenoord in Switzerland if they were to book their place in the final 32.
Defender Alan Maybury, striker Graham Weir and midfielder Michael Stewart came into the Edinburgh side's starting line-up in place of midfielder Joe Hamill and strikers Ramon Pereira and Dennis Wyness.
The altered formation dominated the early play and a Hartley effort forced a good save from Szucs.
Hartley did have the ball in the net in the 14th minute, but the referee ruled that the midfielder had used his hand to control the ball.
Szucs tipped a tremendous McAllister free-kick over the bar.
But only a glaring miss from striker Daniel Tozser from 12 yards out - after good work by Aleksander Bajevski on the left-hand side - let the home side off the hook.
Midfielder Peter Lipcsei then sliced open the home defence and, although Gordon blocked twice at the feet of Rosa, the striker eventually found the net.
A Stewart drive flew a few yards over from the edge of the box and Szucs tipped a deflected drive by Mark De Vries round the post as Hearts tried to battle back into the game.
Stewart threatened immediately after the break and Gabor Gyepes had to clear from the feet of Weir after Szucs spilled the midfielder's low drive.
But Gordon was forced to turn a low drive from Leandro round the post and, from the corner, a header from substitute Thomas Sowunmi struck the bar as Ferencvaros continued to look dangerous.
A Steven Pressley effort flashed wide, but Hearts failed to turn possession into chances as the Hungarians claimed a deserved victory that left both sides out of the competition.
Hearts: Gordon, Neilson, Pressley, Webster, Maybury, Stewart,
Kisnorbo (Wyness 71), Hartley, McAllister (Pereira 62),
de Vries, Weir (McKenna 79). Subs not used: Moilanen, Hamill, Berra, MacFarlane.
Booked: Webster, McKenna.
Ferencvaros: Szucs, Vukmir, Gyepes, Balog, Rosa, Lipcsei,
Zavadszky, Leandro, Tozser (Huszti 50), Bajevski (Takacs 87),
Penksa (Sowunmi 65). Subs not used: Udvaracz, Szkukalek, Somorjai, Bognar.
Booked: Lipcsei, Zavadszky, Tozser, Leandro, Huszti.
Goals: Rosa 30.
Ref: Levan Paniashvili (Georgia).