Philippe Coutinho's moment of brilliance extinguished Manchester United's faint hopes and saw Liverpool through to the Europa League quarter-finals.
Boasting a two-goal advantage from last week's frenetic, one-sided encounter at Anfield, Jurgen Klopp's men made the short journey knowing an away goal at Old Trafford would throw their rivals' comeback hopes off-kilter.
For a while Louis van Gaal's side had looked capable of a memorable comeback, only for Coutinho's magic to kill the mood - and United's hopes of progressing from the last-16 tie - as Liverpool held out for a 1-1 draw on the night, seeing them through 3-1 winners on aggregate.
United were always facing an uphill battle - highlighted by bookmakers pricing them at 7/1 to progress - but started strongly, keeping firm at the back and attacking with panache.
Anthony Martial's penalty as half-time approached gave the home fans hope, which Coutinho snatched away with a mood-changing individual effort, ghosting into the area and clipping past David de Gea.
It all but ended the tie as Liverpool deservedly progressed and narrowed under-pressure Van Gaal's hopes of ending a difficult season with a trophy - and Champions League football.
Unfortunately ugly scenes dominated the final minutes on a night when vile chants about Hillsborough were sometimes audible - songs heard in the first leg at Anfield and looked possible again after a vile banner reading 'Murderers' was displayed on the outskirts of the Manchester.
UEFA may take action this time having not punished United for last week's chants as it was not mentioned in the match officials' reports - the same reason Marouane Fellaini got away with swinging out at Emre Can.
It meant, much to Klopp's chagrin, that the Belgian started and was soon proving a handful, catching Roberto Firmino during an edgy start which took 18 minutes to produce the first shot of note.
When it did Simon Mignolet showed impressive flexibility to claw away Jesse Lingard's close-range header, with the Liverpool goalkeeper no doubt having breathed a sigh of relief when Juan Mata then struck across the face of his goal.
Martial cut in and fired over as United looked to open the scoring, while at the other end last-ditch defending and dithering thwarted the visitors before De Gea - so impressive in the first leg - superbly denied Coutinho.
United quickly capitalised as Nathaniel Clyne, who won the penalty that opened the scoring at Anfield, clumsily clipped Martial in the box, with the France striker slamming home the resulting spot-kick.
Sir Alex Ferguson punched the air in delight - a feeling that Daniel Sturridge nearly abruptly ended with a fizzing free-kick from an acute angle that hitting the crossbar.
Marcos Rojo prodded inches wide as United looked to level the tie on aggregate, only to be hit by a gut punch just before half-time.
The hosts failed to heed Jordan Henderson and Firmino's warning shots, with Coutinho showing exceptional skill to net a crucial away goal.
Running onto the ball wide on the left, he darted towards the area, cut all too easily past Guillermo Varela and, from an acute angle, clipped over De Gea.
The Spanish goalkeeper will have no doubt been frustrated not to keep out an effort which delirious Liverpool fans celebrated throughout half-time.
United attacked strongly after the break - too strongly in the case of Fellaini, who was booked for elbowing Dejan Lovren - but Mignolet's goal was rarely under any serious threat.
Martial missed the target shortly after Sturridge flashed wide at other end, where crowed trouble in the East Stand upper tier saw many miss the final whistle.
It appeared a handful of visiting fans had made it into the home section, with stewards required to break up the brawling.