The Anfield stalemate and subsequent elimination also dealt a blow to their hopes of returning to the competition next season.
Trailing 1-0 from the first leg of their last-16 tie, even a first start for record signing Andy Carroll could not turn things around as the visitors employed every trick in the book to hang on to their advantage.
The Reds have not failed to qualify for Europe since 1999 but having lost the chance of entering as defending Europa League champions, they must now look to their domestic form.
However, they trail fifth-placed Tottenham by six points and have played one match more, and it will take an impressive run of results for them to reel in the Londoners.
With Liverpool low on creativity and even shorter on chances, the watching Reds owner John Henry, who sanctioned the £35million January purchase of Carroll, must have left wondering just how much it costs to buy a goal at this level.
The 22-year-old striker had his moments but very few were in areas in which he could utilise his talent.
Liverpool's defence looked for him at every available opportunity - although sometimes he was not necessarily the best option - and his aerial prowess was soon in evidence.
An early flick-on put Joe Cole through on the left of the area and his shot was parried by Artur back into the path of the former Newcastle striker but his shot was deflected behind.
Carroll headed behind Raul Meireles' corner and it was only set-pieces which really presented him with anything approaching a chance.
Too often the England international was asked to win long diagonal balls 30 yards out with his back to goal.
Liverpool's supposed wide players - in truth the Reds have lacked genuine width for a long time - Cole and Maxi Rodriguez never got themselves into what could be described as threatening positions in which to deliver crosses for Carroll to attack.
The big front man soon discovered everything would not go his way, though, as he was penalised when Paulo Cesar pushed him into Miguel Garcia at another Meireles corner.
Danny Wilson's unfamiliarity at left-back was often exploited by Alan, the goalscorer in Portugal last week, but aside from Rodrigo Lima's drilled 35-yard shot straight at Jose Reina, making his 300th appearance, they were content to soak up the pressure.
However, the methods the Portuguese side used to frustrate their opponents, rolling around on the floor to waste time, required a firm hand and Italian referee Gianluca Rocchi was found desperately wanting as he allowed the game to get away from him before half-time.
Rocchi had no option but to get his cards out soon after the interval when Carroll, frustrated by Paulao's attempts to shepherd the ball out of play, swung a wild tackle at the defender and both players were booked.
Liverpool's first chance of the second half fell to Meireles when Dirk Kuyt crossed to the penalty spot but the Portugal international looped his header well over.
Tension was beginning to grow inside Anfield and the impact of Carroll, watched by England manager Fabio Capello who had arrived from watching Manchester City exit the competition earlier in the evening, was starting to wane.
As the match entered its last quarter Glen Johnson finally put in a dangerous cross but Meireles was just short of reaching it.
One late Carroll header appeared destined to find the net from a Meireles corner only for Kuyt to get in the way in the six-yard area.
That just about summed up his - and Liverpool's - night as they went out of the competition despite not losing a home match in Europe all season.