Both sides had good chances in the opening half hour from Jesse Lingard and Philippe Coutinho before Anthony Martial converted the penalty. Coutinho cancelled that out on the stroke of half-time.
It expected to be a heated contest at Old Trafford as United looked to get themselves back into the tie, but the first opportunity didn't arise until the 20th minute.
Simon Mignolet, who didn't have much to do in the first leg at Anfield, was called upon here as Lingard looked to turn his close-range header in, the Belgian managed to tip it over the crossbar.
It was then the turn of David De Gea who did well with the double save just before half-hour mark.
Coutinho had a very similar opportunity from close range to that of Lingard but De Gea was down to tip it away from the bottom corner before smothering the ball to prevent the Robert Firmino follow-up.
Martial draws first blood
It was United who were handed the perfect opportunity to grab the crucial opening goal when referee Milorad Mazic pointed to the spot after Nathaniel Clyne brought down Martial in the area.
It was the Frenchman who stepped up to take the spot kick after earning it, and he made no mistake as he cannoned it into the back of net.
Daniel Sturridge almost had a instant reply as he attempted to curl the ball around the wall from a set-piece of his own, but the Englishman could only find the woodwork.
Jordan Henderson had a great chance to equalise in the 42nd minute, and Coutinho, who provided the opportunity, was much more clinical on the stroke of half-time to equalise.
It was a great piece of individual magic from the Brazilian as he managed to glide to the byline, leaving Guillermo Varela in his wake, and wait for De Gea to commit himself before chipping it over the Spaniard to effectively end the tie.
Van Gaal's side came out for the second period knowing they had a mountain to climb if they were to reach the last eight but only managed to muster up a few half-chances.
The first came from a combination of youth as Lingard played it through to new star Rashford who got his shot away on the edge of the area but went wide of Mignolet's post.
Fellaini followed that up on the hour mark as he was presented with a header on a plate after a lovely ball from Michael Carrick but his header could only find the hands of his compatriot Mignolet.
The Belgian had an even better chance in the 72nd minute when Antonio Valencia pulled it back into his path but he hit the shot all wrong as it rocketed over the crossbar and it didn't look to be United's day.
Seeing it off
As the match entered it's closing stages Jurgen Klopp's side looked to seal the win and some good chances to finish off United but De Gea was on hand to give United a glimmer of hope.
He produced two great saves with 11 minutes left to play tipping the first effort over the crossbar from Coutinho's effort from inside the 'D'. The second followed almost instantaneously as Mamadou Sakho headed into the path of Firmino who forced De Gea to sprawl across to get a hand to it but it was all in vain as the offside flag was raised.
The final shot of the night fell to the substitute Origi who hit a speculative effort at De Gea who was forced to parry which nearly fell to Joe Allen but he couldn't put a final nail in the already doomed Manchester United coffin.