(CNN) -- Liverpool have crashed out of the Champions League despite a narrow 1-0 Group E win over Debrecen in Hungary.
The English Premier League side, who last won the competition in 2005, needed to win to give themselves any chance of reaching the last 16, and they achieved that in a tight match.
However, Fiorentina's 1-0 home win over Lyon rendered the result irrelevant and means Liverpool willfinish third in the group and drop down into the Europa League.
French striker David Ngog scored Liverpool's early winner, but Rafael Benitez's side struggled to produce a convincing display, the damage having been done in two games against Lyon that cost them four points.
The road to Madrid, where the Champions League final is due to be played, now becomes the road to Hamburg -- where the new Europa League will reach its conclusion -- with Liverpool in with the likes of Valencia, Benfica and city rivals Everton.
Liverpool could not have asked for a better start to settle any nerves as they went ahead after just four minutes with the Hungarians having barely touched the ball.
Fabio Aurelio's cross from the right curled to the far post for Jamie Carragher to head back across goal, where Ngog guided the ball with the outside of his right foot in off the keeper and the far post.
Twice Dirk Kuyt narrowly failed to get on the end of close-range crosses as Liverpool sought a second, while Carragher headed an Aurelio corner just wide, but on a night where only nine goals were scored in eight Champions League matches, N'gog's goal was to prove the only one of the game.
Meanwhile, Fiorentina booked their place in the last 16 with a narrow victory over Lyon in Florence.
A Juan Vargas penalty in the 28th minute was enough to see the Viola successfully navigate their way through the group stages for the first time in their history, joining Lyon in the knockout phase.
A sell-out crowd of over 35,000 witnessed Fiorentina's historic achievement as they also became the first Italian representatives to qualify for the knockout round this year.
Mario Santana almost put them ahead in the 11th minute with a fine control and volley from the edge of the penalty area which struck Hugo Lloris' right-hand post.
The danger was not over, however, as Alberto Gilardino met the resulting corner with a header which brushed the opposite post.
Fiorentina continued to probe and they were rewarded in the 27th minute when Lyon defender Aly Cissokho unnecessarily hacked Marco Marchionni down inside the area to give away a penalty.
With first-choice penalty takers Adrian Mutu and Stevan Jovetic both injured, Vargas took responsibility to score his first goal of the season at the Stadio Artemio Franchi, placing his penalty into the top left-hand corner of the goal.
The results mean Fiorentina qualify with 12 points, while Lyon are on 10 and can still finish as group leaders.
Liverpool are third on seven points but cannot finish above Lyon on their head-to-head records, while Debrecen are still without a point from their five matches.
Meanwhile, Liverpool's compatriots Arsenal made no mistake in their quest to secure qualification, cruising to a 2-0 home win over Standard Liege to finish top of Group H.
Arsenal came into the match needing just a draw to qualify and on the back of a surprise 1-0 Premier League defeat at Sunderland -- and they were always in control against their Belgian opponents.
First-half goals from Samir Nasri and Denilson set the Gunners on their way to the knockout stage for a 10th campaign in succession.
The hosts took the lead on 35 minutes when defender Landry Mulemo slipped on the edge of the penalty area, allowing Nasri to lash home a half-volley.
Then, in first-half stoppage time, Denilson crashed home the second goal from 30 yard out, the ball swerving through the air to give goalkeeper Sinan Bolat no chance.
The visitors' poor night was complete late on, when Mehdi Carcela-Gonzalez pushed his forehead into the face of Arsenal captain Cesc Fabregas and was shown a straight red card.
Despite the defeat, Standard Liege can still reach the knockout stages after AZ Alkmaar and Olympiakos drew 0-0 in the other group match.
The results mean Arsenal top the group with 13 points, Olympiakos are on seven and Liege on four.
However, if Arsenal win at Olympiakos in the final group match -- and Liege defeat AZ Alkmaar at home, the Belgian side will snatch second place on head-to-head records.