Match ends, Manchester United 1, Liverpool 1.
Philippe Coutinho's brilliant solo goal killed Manchester United's hopes of a Europa League comeback and sent Liverpool into the last eight in comfort at Old Trafford.
Jurgen Klopp's side were protecting a two-goal lead from the first leg at Anfield but Anthony Martial's 32nd-minute penalty offered United brief hope after he was fouled by Nathaniel Clyne.
Jesse Lingard and Juan Mata had missed clear opportunities for United but Liverpool were also a threat as David de Gea saved superbly from Coutinho while Daniel Sturridge hit the bar with a free-kick and Jordan Henderson missed an open goal.
The away goal Liverpool threatened, and which left United needing four on the night, came right on half-time when Coutinho beat Guillermo Varela with ease before lifting a near-post finish past De Gea.
It ended the game as a contest with United unable to rouse themselves again as Liverpool completed the formalities to win the first European meeting between the two clubs.
Sadly, the final stages of the match were marred by clashes between supporters near the Liverpool corner of the ground - and it remains to be seen if Uefa take any action.
Too little too late for Man Utd
Manchester United, in the first half at least, played with a pace, and showed a spirit and endeavour, that has rarely been displayed at Old Trafford this season - but the task was too much.
And that was down to the lamentable performance at Anfield last Thursday when Liverpool dominated every facet of the first leg to secure a two-goal lead that could actually have been much more.
It left United vulnerable to one goal from Liverpool that would leave them needing four, and so it proved when Coutinho's brilliant dribble provided that crucial away strike.
United, from that point on, looked like the team they have been for most of this season - pedestrian, uninspired and struggling to illuminate an Old Trafford stage that has suffered much this season.
And there was suffering in hearing songs of celebration from 3,000 Liverpool fans tucked away in one corner of the stadium as their arch-rivals celebrated victory in the first European meeting between the two sides and a place in the last eight of the Europa League.
It was also further evidence that, for all those who still feel United might be better off without Wayne Rooney, that this team still misses him very badly as he was reduced to watching the game from the stands with his son as the England captain recovers from a knee injury.
Can Liverpool go all the way?
Liverpool are progressing rapidly under Klopp, who is putting his imprint on the side six months after he succeeded sacked Brendan Rodgers.
And some of the old European anticipation will be rising at Anfield as they contemplate Friday's quarter-final draw.
Time for some perspective, though, and a warning that Klopp is embarking on a rebuilding programme that will not be a quick fix.
There are still some tasty potential opponents lying in wait, none more so than Klopp's old club Borussia Dortmund, tournament favourites and impressive winners against Tottenham over two legs.
The notion of Klopp facing Dortmund is an enticing one and he would be guaranteed a rapturous reception at the club he took to two Bundesliga titles and the Champions League final.
And that is before you even get to Sevilla, winners in the past two seasons and who cruised into the last eight against FC Basel.
Young guns offer hope for Man Utd
This was another disappointing night in a disappointing season for Manchester United - and there will be extra pain in losing a European tie to such fierce adversaries.
If there is some comfort, it can be gained in the performances of young strike duo Marcus Rashford and Anthony Martial.
Rashford, just 18, never gave up at any point, chasing lost causes endlessly while trying to provide United's lost spark.
Martial, 20, tormented Clyne with his pace in the first half to earn - and score - the penalty that gave United some hope.
They are small crumbs of comfort on a miserable night for United but they need all the hope they can get after this.
'Punches and seats thrown'
Liverpool's on-field celebrations were marred by trouble in the stands in the closing minutes and after the final whistle.
BBC Radio 5 live commentator Ian Dennis had a clear view of the disturbances in the crowd and said he saw fighting and seats being ripped out.
"Punches are being traded by rival supporters," he said.
"I have seen three red seats from the Liverpool section being thrown into the Manchester United fans.
"There are about 10 Liverpool supporters sat on the front row of the top tier and there is a human wall of police officers in luminous clothing protecting the Liverpool fans sat in the Manchester United end.
"Uefa will take action - mark my words."
Man of the match - Philippe Coutinho
What the managers said
Manchester United boss Louis van Gaal: "I am not angry, I am not frustrated. I was very proud of my players.
"They gave everything and I was very pleased the fans recognised that. They applauded after the match despite us being out - that was remarkable I think.
"I hope Man City is the catalyst. We have to beat City and we have a chance to still qualify (for the Champions League)."
Liverpool manager Jurgen Klopp: "They got a penalty - it was a penalty - and Phil had a genius moment.
"I love it two minutes before half-time because he twice reacted quicker than his opponent.
"It was the most unexpected that he could do in a situation like this."
Manchester United will travel to the Etihad for a Premier League derby against neighbours Manchester City on Sunday, the same day as Liverpool play Southampton.
The stats you need to know
- Anthony Martial has scored in back-to-back games for Manchester United for the first time since September 2015
- Coutinho has equalled his best goals tally in a season for Liverpool (8) in all competitions, a feat he achieved last season
- The Red Devils have been eliminated from the Europa League at the last-16 stage in both of their appearances in the competition (also 2011/12 versus Athletic Bilbao)
- Liverpool have reached the quarter-final of a European competition for the first time since 2009/10, when they faced Benfica in the Europa League on their way to the semi-finals
- Jurgen Klopp is the first Liverpool manager since Graeme Souness in October 1991 to avoid defeat in his first game at Old Trafford as Reds boss
- 1de Gea
- 30VarelaSubstituted forA Valenciaat 45'minutes
- 17BlindBooked at 33mins
- 5RojoSubstituted forDarmianat 62'minutes
- 16CarrickSubstituted forSchweinsteigerat 70'minutesBooked at 80mins
- 27FellainiBooked at 50mins
- 25A Valencia
- 2ClyneBooked at 51mins
- 14HendersonSubstituted forAllenat 71'minutes
- 10CoutinhoBooked at 42mins
- 11FirminoBooked at 83minsSubstituted forBentekeat 85'minutes
- 15SturridgeSubstituted forOrigiat 68'minutes
- 4K Touré
- 44B Smith
- Milorad Mazic
- Shots on Target
Second Half ends, Manchester United 1, Liverpool 1.
Attempt saved. Divock Origi (Liverpool) right footed shot from the left side of the box is saved in the bottom right corner. Assisted by Christian Benteke.
Attempt saved. Divock Origi (Liverpool) right footed shot from outside the box is saved in the centre of the goal. Assisted by Christian Benteke.
Corner, Liverpool. Conceded by Matteo Darmian.
Attempt missed. Jesse Lingard (Manchester United) right footed shot from outside the box misses to the right. Assisted by Marouane Fellaini following a corner.
Attempt missed. Matteo Darmian (Manchester United) right footed shot from the centre of the box misses to the left following a corner.
Attempt missed. Bastian Schweinsteiger (Manchester United) right footed shot from the left side of the box misses to the right following a corner.
Corner, Manchester United. Conceded by James Milner.
Substitution, Liverpool. Christian Benteke replaces Roberto Firmino.
Attempt missed. Daley Blind (Manchester United) left footed shot from outside the box misses to the left following a set piece situation.
Roberto Firmino (Liverpool) is shown the yellow card for a bad foul.
Jesse Lingard (Manchester United) wins a free kick in the attacking half.
Foul by Roberto Firmino (Liverpool).
Foul by Chris Smalling (Manchester United).
Divock Origi (Liverpool) wins a free kick in the defensive half.
Attempt missed. Philippe Coutinho (Liverpool) right footed shot from outside the box is too high from a direct free kick.
Bastian Schweinsteiger (Manchester United) is shown the yellow card for a bad foul.
Foul by Bastian Schweinsteiger (Manchester United).
Adam Lallana (Liverpool) wins a free kick in the attacking half.