The Reds, protecting a two-goal lead from last week's first leg on Merseyside, got the job done thanks to a 1-1 draw on the night to seal a 3-1 progression over the two games.
A fourth game without a win in all competitions for the Red Devils essentially ends their season, with just a top-four place in the Premier League left to fight for ahead of a trip to face near neighbours Manchester City at the weekend.
For Liverpool, on the other hand, their impressive recent run continues and they have a last-eight tie to look forward to when the draw for the next round is made on Friday morning.
The opening quarter of the match passed by without too much incident, but it was the home side who were quickest out of the blocks when putting together some positive attacking moves.
From the resulting set piece Juan Mata volleyed wide of the far post from eight yards out, yet it was not all United's way as Coutinho was to be denied by a fine David de Gea stop at the other end just moments later.
De Gea proved a key figure in the opening fixture between these two English heavyweights, and so it proved again with arguably the pick of his saves to thwart Coutinho's strike from the edge of the box.
There was a sense that the tide was beginning to turn in the Reds' favour when Roberto Firmino, a player who had scored three in his last three outings, also tested the Spanish keeper with a far more routine stop.
The big breakthrough of the half finally arrived a little over half-an-hour in, as Martial was brought down by Nathaniel Clyne inside the box before sending Mignolet the wrong way from the spot.
Plenty more drama was still to arrive prior to the interval, however, with Daniel Sturridge sending a free kick back off the crossbar and Jordan Henderson somehow failing to find the target when played through on goal by Coutinho's smart pass.
Marcos Rojo's miss from close range at the end of a Marouane Fellaini knockdown looked to have been the last attacking play of the half, but right on the stroke of the interval Coutinho waltzed past Guillermo Varela and tucked the ball beyond the reach of De Gea with a delicate flicked finish at the front post.
That was Coutinho's first European goal for the club at the 14th attempt, and it changed the mood of the game completely as the atmosphere inside the Theatre of Dreams audibly reduced by a few notches.
The big talking point in a quiet opening period to the second half came when Fellaini caught Dejan Lovren with a flailing elbow, which appeared to have just an element of intent to it.
Referee Milorad Mazic disagreed, showing the United forward - who came close shortly after the incident with a header down Mignolet's throat - just a yellow card.
As the less lively second half wore on, the theme of the game became clear - Liverpool breaking into gaps that opened up at the back without really capitalising on their openings.
United were still putting together some nice passing moves, something which offers just a glimmer of hope for the remainder of the campaign, but there was a sense that supporters were beginning to lose patience once Fellaini scooped a shot over the bar 20 minutes from time.
There was a good chance for Coutinho to wrap up the game for good late on when working a yard of space inside the area, yet he could only pick out De Gea rather than the gaping goal.
All that was left was for the Serbian official to blow his whistle for full time, signalling a famous win for Liverpool in the 196th meeting between English football's two most successful sides.