The Reds' fate was sealed in Florence, where already qualified Lyon sunk to a 1-0 defeat to Fiorentina.
Perhaps the only positive to emerge from the miserable night for Liverpool manager Rafael Benitez was the assured performance of his players, who will be left scratching their heads as to how they did not win the game by three or four goals after a dominant display which kept the Reds camped in the final third.
The match-winner was young Frenchman David N'Gog, the understudy to injured striker Fernando Torres.
N'Gog flicked the ball home from two yards out after Jamie Carragher headed Fabio Aurelio's set-piece delivery back across goal to provide a cushion in the game in just the fourth minute.
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From then on, long periods of the match resembled men against boys, as the likes of Javier Mascherano, Glen Johnson, Dirk Kuyt and Steven Gerrard passed the ball around the Hungarians with ease.
But the rest of the half saw little goalmouth action, with N'Gog producing the only effort of note with a stinging 20-yard attempt which was superbly saved by goalkeeper Vukasin Poleksic, the home side's star man of the night.
The second half was played to the same script, as the Reds were presented with the lion's share of possession seemingly with little effort.
It was Gerrard who came closest to doubling the lead on the hour. The captain was presented with a chance he would normally bury from 12 yards but saw his left-footed effort saved well low to Poleksic's right.
Liverpool were almost put out of their misery in stoppage time. Lively substitute Tibor Dombi should have sealed a historic point for the minnows by latching on to a loose ball 10 yards out but fired straight at Pepe Reina.
Reality quickly dawned on a miserable Liverpool dressing room as news filtered through of Fiorentina's success over Lyon, as the Reds failed to get out of the group stage in the Champions League for the first time in seven years.