Ole Gunnar Solskjaer's side went into the tie as the underdogs but more than held their own.
They actually threatened an upset when Marcus Rashford netted in the first half.
But while Rashford would get all the plaudits, the goal was made by Daniel James' superb work down the right-hand side.
The Welshman, who was experiencing the fierce United vs Liverpool rivalry for the first time, sprinted down the wing and put in a delicious ball for his teammate.
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Rashford could not miss and he had the easy task of putting it in the net.
But, all-in-all, it was a rough game for James.
The former Swansea player's pace means he is arguably United's most dangerous player.
That often makes him a target for opponents, who want to rough him up and try and limit his influence on the game.
And on Sunday that was the same again. James was on the other end of some quite brutal challenges from Liverpool players.
Five minutes into the second half, the Welshman broke away with the ball, only to be taken down by Virgil van Dijk.
The Dutchman got the ball but then made sure he also took out his man.
Watch Van Dijk's challenge on James below:
Play was eventually stopped as James was winded on the floor.
After a few minutes he got up and played on, only to be taken down by Fabinho in what was an even stronger challenge.
Watch Fabinho's challenge on James below:
Again, the Brazilian midfielder got the ball but then made sure he took out his man.
Joel Matip eventually kicked the ball out as James got treatment.
The 21-year-old got up once again and stayed on the pitch for the rest of the game.
But he may feel the effects of those challenges for a few days, at least.News Now - Sport News