Jurgen Klopp celebrated his first win at Old Trafford as Liverpool came from behind to beat Manchester United and boost their chances of a Champions League place.
In a thrilling game played against the backdrop of noisy protests against the United ownership outside the ground, United struck first through Bruno Fernandes' deflected opener.
But Liverpool, who had seen one of their buses prevented from picking team members up before the game after being blocked in the street, hit back through Diogo Jota's instinctive effort.
Roberto Firmino, who had only scored once since his two-goal haul at Crystal Palace in December, then put his side on top with a well-taken double either side of half-time.
Marcus Rashford pulled one back for United midway through the second half but Mohamed Salah wrapped up the points with a late breakaway strike for his 21st Premier League goal of the season.
That moves him level with Harry Kane in the Golden Boot standings and condemns Ole Gunnar Solskjaer's side to a sixth home defeat of the season.
The result takes Liverpool fifth in the Premier League. If they win at West Brom on Sunday they will close to within one and three points respectively of FA Cup finalists Chelsea and Leicester, who still have to play each other at Stamford Bridge on 18 May.
Liverpool make light of penalty controversy
Brendan Rodgers was in charge in March 2014, the last time Liverpool beat United in an away game.
Jordan Henderson's groin injury meant there was no-one left from that night in the visitors' ranks, and Klopp celebrated his debut United away win with his double fist-pump as Salah scored.
Salah might have had the chance to get on the scoresheet much earlier in the game had referee Anthony Taylor not consulted the pitch side monitor and changed his decision to give a penalty for Eric Bailly's tackle on Nat Phillips.
Knowing victory was needed to maintain a realistic top-four chance, Klopp appeared distinctly unimpressed by the incident. The Ivorian defender certainly played the ball first but the force behind the challenge sent his defender flying.
In a game that started to the sound of a police helicopter whirring overhead, Liverpool did not have to wait long for another chance.
Jota's finish was excellent but United's defence stood off Phillips after a corner, allowing him to run and turn before his shot spun to the feet of his Portuguese team-mate.
Trent Alexander-Arnold's superb free-kick delivery to the far post in first-half stoppage time invited Firmino's header to put Liverpool ahead - and was a timely reminder of his ability with England boss Gareth Southgate watching from the stands.
United struggle without Maguire
This was one of those nights that made you wonder how United have managed to get themselves in line for a second-place finish.
Unlike Tuesday's defeat by Leicester, this could not be put down to a weakened team.
The hosts even had a helping had for their opening goal, as Phillips turned home a Fernandes effort that may just have been curling in at the far post.
Liverpool's first two goals exposed long-existing frailties defending set-pieces. Their third was terrible defensively, as Fred giving the ball away with a blind pass to the wing from inside his own box.
Dean Henderson then spilled Alexander-Arnold's fierce shot straight to Firmino, who applied the first-time finish.
Another misplaced Fred pass began the counter-attack that led to Jota firing against the post. Little wonder Harry Maguire, sat in the stands nursing an ankle injury that threatens to keep him out of the Europa League final with Villarreal on 26 May, bellowed encouragement from the sidelines.
The flowing move that led to Rashford's goal on 68 minutes showed United's attacking ability but, without Maguire, they struggle badly at the back at Salah's pace exposed them again at the end to finish off any hope of a comeback.
Angry United fans make voices heard again
Addressing the protests before the game that Greater Manchester Police revealed had led to two arrests, a club official said the Glazer family were committed to communicating to supporters in a different way.
Quite simply, many of those fans do not believe the message.
Co-chairman Joel Glazer is due to speak at a Fans Forum for the first time after the season has finished. Before that though, United must get through their final home game of the campaign, against Fulham on 18 May, when 10,000 fans will be in attendance.
It is impossible to conceive the barriers that have been erected outside the stadium will remain with so many people about. It is fairly obvious the huge numbers of police will be back, although there is nothing they can do to stop the anti-Glazer chants that seem almost inevitable.
Salah joins exclusive club - the stats
Manchester United have lost two home games in the space of three days for the first time since April 1958.
United have conceded 27 home goals in the Premier League this season, their most in a single league campaign since 1969-70 (also 27).
Liverpool are unbeaten in seven Premier League games (W5 D2), the longest current run without defeat in the competition. The Reds had lost six of their seven league games prior to this run.
A total of 33% of United's Premier League goals conceded this season have been from set piece situations (14/42, excluding penalties), the highest ratio in the division.
This was the second Premier League home game this season that United found themselves behind at half-time having opened the scoring (also v Spurs) - this had only happened once in the previous 28 Premier League campaigns (v Liverpool in 2008-09).
In his 200th appearance for Liverpool in all competitions, Salah netted his 124th goal for the Reds - only Roger Hunt (133) and Gordon Hodgson (125) scored more for the club in their first 200 games.
Firmino scored his 17th headed goal in the Premier League for Liverpool - only Robbie Fowler (21) has scored more headers for the Reds in the competition.
Rashford has scored five goals in his last five appearances against Liverpool, with Thursday's strike his 50th at Old Trafford in all competitions for United.
Making his 50th Premier League appearance, Fernandes' opener was his 44th league goal involvement for United (26 goals, 18 assists). Only Eric Cantona (46) and Robin van Persie (45) have been involved in more in their first 50 for the club in the competition.
Manchester United are next in action against Fulham, which is due to be the first game at Old Trafford in front of fans since March last year, on Tuesday, 18 May (18:00 BST). Liverpool go to relegated West Brom on Sunday, 16 May (16:30 BST).