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Jurgen Klopp‘s side went in front during the early stages through Virgil van Dijk, only to have two goals ruled out later in the first half.
However, despite United creating several chances of their own, the Merseyside giants sealed the win in the final seconds through Mohamed Salah, who finally ended his lengthy streak without a goal against United.
While Liverpool move further clear of Manchester City and Leicester City, United remain in fifth position, five points adrift of Chelsea in the race for a Champions League spot.
With Marcus Rashford sidelined with a back injury, Ole Gunnar Solskjaer opted to hand a recall to Luke Shaw, leading to Brandon Williams being used further down the left.
However, while United held their own during the opening stages of the contest, Liverpool were still able to take the lead in the opening quarter-of-an-hour through Virgil van Dijk, who rose above the defence to power a header into the top corner.
Despite chances remaining few and far between, Liverpool stayed in the ascendancy, and they looked to have doubled their advantage when Roberto Firmino curled a delightful shot into the far corner from 14 yards.
VAR ruled otherwise, however, with it is being deemed that Van Dijk had unfairly challenged David de Gea in the build-up to the goal.
While it appeared harsh on the home side, they should have made it irrelevant minutes later, only for Firmino to somehow divert a half-volley wide of the post from eight yards after connecting with Mohamed Salah’s cross.
Liverpool soon had another goal disallowed, although the assistant referee correctly adjudged Georginio Wijnaldum to be offside as the Dutchman coolly found the far corner from an acute angle.
United, for the most part, had offered little in the final third, but that changed towards the end of the first half as Anthony Martial fired marginally wide from the edge of the area.
Andreas Pereira should have perhaps done better at the far post when attempting to meet Aaron Wan-Bissaka‘s cutback, before the Brazilian tested compatriot Alisson with a long-range drive.
Nevertheless, Liverpool should have ended the opening 45 minutes two goals to the good, with Sadio Mane firing too close to De Gea when being sent through on goal.
Within two minutes of the restart, the home side created yet another opportunity as Andrew Robertson teed Salah up for what appeared to be a tap-in, only to witness his teammate somehow scuff a shot wide of the target.
Moments later, Jordan Henderson rattled the post with well-struck effort from 20 yards, before Alex Oxlade-Chamberlain saw a long-range strike well held by De Gea.
Despite being second-best, United remained a threat on the break, and Fred should have done better after a powerful run forward instead of sending a tame attempt off target.
Martial was then guilty of squandering an opening inside the area, volleying well over the crossbar when it seemed easier to find the far corner.
As the second half progressed, the clear-cut chances began to dry up, but both clubs continued to push for the second goal which an enthralling contest deserved.
The two managers opted to make double alterations in order to give themselves an extra edge, although the introduction of Fabinho indicated that Jurgen Klopp was happy to settle for 1-0.
United, as expected, pushed for a leveller, but Liverpool were able to confirm a 21st win from 22 outings in the final minute through Salah, who slotted a low shot past De Gea after being played in by a long ball from Alisson.
The goal was the least that Liverpool deserved after incessant pressure earlier in the match came to nothing, although United may feel that this was a missed opportunity of their own after finding a way to remain in the contest throughout the 90 minutes.
LIVERPOOL XI: Alisson; Alexander-Arnold, Gomez, Van Dijk, Robertson; Oxlade-Chamberlain, Henderson, Wijnaldum; Salah, Firmino, Mane
MANCHESTER UNITED XI: De Gea; Wan-Bissaka, Lindelof, Maguire, Shaw; Matic, Fred; James, Pereira, Williams; Martial