Harry Maguire’s 84th-minute header gave United a colossal win, their first at Anfield in almost a decade, and secured three points in successive games for the first time under Ruben Amorim.
Liverpool, meanwhile, find themselves having lost four times in a row, and now sit four points behind league-leaders Arsenal.
United made a terrific start, Amad Diallo curving a lovely pass into the path of Bryan Mbeumo, who ran off Virgil van Dijk and expertly finished past Giorgi Mamardashvili with just 62 seconds on the clock.
The goal was not without controversy, though. Earlier in the move, Alexis Mac Allister was caught by Van Dijk and went down with a head injury, but referee Michael Oliver allowed the game to continue and the visitors crafted a fine opener.
The tempo did not let up. Both sides created chances – Cody Gakpo twice hit the post while Senne Lammens saved well from Alexander Isak while, for United, Bruno Fernandes also struck the woodwork.
Liverpool were dominant after the break as United sat back, and it was no surprise when the pressure eventually told. Gakpo tapped home on 78 minutes after good initial work from Van Dijk and a pinpoint cross from Federico Chiesa.
The visitors, though, offered a constant threat on the counter and after winning a corner down the left, first Mbeumo’s shot was blocked before Bruno Fernandes’ volleyed a brilliant first-time cross for Maguire, who cushioned a header into the corner to seal a huge victory.