The Reds advanced 3-1 on aggregate after their 2-0 first leg win at Anfield.
United’s exit leaves embattled manager Louis Van Gaal with only the English FA Cup to concentrate on for silverware this season.
The Red Devils’ are currently sixth in the Premier League and will have to make up a four-point margin to overhaul neighbours Manchester City to claim the final Champions League berth for next season.
United played their way into the tie when Anthony Martial converted from the penalty spot in the 32nd minute, but all hopes evaporated when Brazilian Philippe Coutinho equalised with a crucial away goal late in the first half to effectively kill the tie.
Liverpool manager Juergen Klopp hailed Coutinho’s wonder goal, scored on the stroke of halftime by dribbling into the box and applying a deft chip from an acute angle.
“Phil had a genius moment, wonderful,” Klopp said, “and it decided the game.”
United’s only possible route to next season’s Champions League rests solely through a top-four finish in the Premier League, which is currently looking unlikely with the team in sixth place with nine games left.
Sunday’s derby against Manchester City, who are in fourth place and four points ahead of United, takes on even greater significance for United and their under-pressure manager Louis van Gaal.
Progress for United and the Dutchman in the Europa League was always in doubt after the first leg last week that Liverpool dominated and won 2-0 at Anfield.
It meant United had to come out and attack at Old Trafford, leaving the home side susceptible to counterattacks from Liverpool’s lively forwards.
From one of those breaks, Coutinho collected the ball on the left wing, drifted past right back Guillermo Varela, closed in on goal and executed an extravagant chip that beat David de Gea at his near post.
It punctured an atmosphere that had gone up a notch inside Old Trafford when Martial gave United the lead from the penalty spot, after he was tripped by Nathaniel Clyne.
United never looked like scoring the three goals it needed in the second half, which finished amid troubling scenes in one of the stands as a small number of United and Liverpool fans threw punches at each other before stewards intervened.
“We have played like Liverpool did in the home match against us,” Van Gaal said, “… but they scored the away goal.
“That was the difference. In the second half, you try to raise the players out of disappointment, but you know and the fans know you need to score three goals.”
Tottenham Hotspur’s run in the competition also came to an end when they were beaten 2-1 by Borussia Dortmund.
Dortmund, who were previously coached by current Liverpool manager Juergen Klopp, went through 5-1 on aggregate.
Spurs’ defeat will allow the London outfit to concentrate on their EPL title assault where they are second behind Leicester City and in prime position to secure one of the four Champions League spots on offer.
Meanwhile Athletic Bilbao, Villarreal and Sparta Prague reached the quarter-finals of the Europa League at the expense of Valencia, Bayer Leverkusen and Lazio.
Bilbao lost 2-1 to Valencia, who started with Socceroo Mat Ryan in goal, but squeaked through on the away goals rule, with a 2-2 aggregate.
Valencia’s exit from Europe’s second-tier competition heaps more pressure on under-fire English manager Gary Neville, who was sent off for vehemently protesting against Bilbao’s controversial killer goal.
Valencia swept into a promising 2-0 lead but conceded the vital away goal to Bilbao’s ageless striker Aritz Aduriz.
Santi Mina opened the scoring for Valencia in the 13th minute after good work down the left by the impressive Jose Gaya.
Defender Aderlan Santos made it 2-0 eight minutes before half time by heading in an excellent centre from Gaya.
Bilbao dominated the second half and tournament top scorer Aduriz, 35, saved the Basques from elimination by beating Ryan 15 minutes from time, despite what seemed to be a handball by Markel Susaeta earlier in the move.
“I never like to put the blame on the referee, but when he sees [on television] what occurred, he will be disappointed with himself,” Neville said.
“I didn’t insult him at all. I just said that it was handball. This was a very unfair match. We should have won by four or five goals. You need a bit of luck in football, and we didn’t have that tonight.”
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