Anthony Martial gave United hope with a first-half penalty but Coutinho equalised just before the break.
And that ended the game as a contest as the Red Devils were left needing three more goals to overturn their north-west rivals.
Liverpool will now go into the hat for the quarter finals – along with Borussia Dortmund, who beat Tottenham – while United head for the crucial Manchester derby on Sunday.
The tempo of the game was high from the offset as it quickly became a game of end-to-end football without any real clear-cut chances.
The first real effort came from Jesse Lingard, who leapt highest but aimed his header from Marcos Rojo’s whipped cross too close to Simon Mignolet, who made a fantastic save.
It was David De Gea’s turn to pull off a phenomenal save next as he got down brilliantly to push away a fierce Coutinho shot from the edge of the area.
Martial and Marcus Rashford began to stretch the Liverpool defence as United were on top and it was the former whose dangerous run saw Nathaniel Clyne stick out a leg and bring him down in area.
The Frenchman picked himself up and dusted himself off to lash home the penalty to grab a goal back in the tie.
Liverpool nearly hit straight back but Daniel Sturridge’s whipped free-kick cannoned back off De Gea’s crossbar.
The Reds did get their goal with the last kick of the half as Coutinho ghosted past Guillermo Varela too easily before classily dinking the ball over De Gea to restore his side’s two-goal aggregate lead.
Marouane Fellaini was lucky not to see red in the first leg for an elbow and he was fortunate again after half time as he thrust his elbow into the back of Dejan Lovren’s head.
Needing three goals to win the tie now extra time is out of the question, Louis van Gaal’s three substitutions would not have pleased United fans as both full-backs and Michael Carrickwere replaced by Matteo Darmian, Antonio Valencia and Bastian Schweinsteiger.
Martial and Fellaini both had chances to grab goals but both fired high and wide when they should have done better.
And that was it for the Red Devils’ Europa League season meaning Manchester United were knocked out of Europe by a team from their own country for the first time in their history.