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Ole Gunnar Solskjaer's side took the lead early through Bruno Fernandes, but were pegged back before half-time as Diogo Jota and Roberto Firmino got on the scoresheet.
It looked like the visitor had things wrapped up early in the second period as Firmino headed in his second only for Marcus Rashford to set up a nervy finish as he pulled one back with 20 minutes remaining.
But the visitor rounded things off late on through Mohamed Salah to put pressure firmly back on Chelsea and Leicester City in the chase for top four.
Liverpool saw penalty claims for handball against Eric Bailly waved away, while Edinson Cavani shot wide from a wayward Alisson pass in a busy opening.
It was United who struck first, Fernandes afforded plenty of space to curl the ball in off Nathaniel Phillips.
Bailly had another escape when VAR showed he made a clean tackle on Phillips after referee Anthony Taylor had initially pointed for a Liverpool penalty.
The visitor was level, though, when Phillips collected the ball from a corner and made space for a shot that Jota expertly diverted home.
Liverpool then deservedly went in front on the stroke of half-time when Firmino headed home Trent Alexander-Arnold's free-kick at the back post.
And it doubled their lead less than a minute after the restart as Dean Henderson spilled an Alexander-Arnold shot to hand Firmino a simple close-range finish.
The visitor could have wrapped it up by the hour mark, with Jota cracking the post and Alexander-Arnold shooting at Henderson from similar positions inside the box in quick succession.
It looked like they would rue that miss when Rashford slightly scuffed a finish home after a brilliantly slick interchange between United's forwards.
But United could not turn its dominance of the final 20 minutes into a goal that would have denied its rival a first Old Trafford win since 2014, which was wrapped up in the final minute of normal time by a cool Salah finish.