Substitute Bruno Fernandes scored the winner with 12 minutes left after Edinson Cavani had won a dubious free-kick on the edge of the area.
United might have felt they were playing Liverpool at the best possible time after the Reds' defeat to Burnley on Thursday, a result which ended their three-and-a-half year unbeaten home run and led to serious questions were being asked about the champions' decline.
Yet there were no signs of that malaise when they opened the scoring at Old Trafford after just 18 minutes through Mohamed Salah.
The Egyptian latched onto a brilliant pass from Roberto Firmino to put the visitors 1-0 up, but Mason Greenwood finished off a rapid counter attack by slotting past Alisson with a cool finish.
Marcus Rashford had set up Greenwood with an incredible cross-field pass, but he then turned goalscorer within minutes of the second half beginning, breaking through after Rhys Williams completely mistimed his interception.
It was a horrible moment for the 19-year-old and one which underlined why Jurgen Klopp is so keen to sign another centre-back in Virgil van Dijk's absence. In recent weeks, he has even preferred to use Jordan Henderson in defence rather than trust in his youngsters.
All wasn't lost for Liverpool, though, as Cavani gave the ball away inside his own half and within seconds, Salah had made it 2-2.
The introduction of Sadio Mane suggested Klopp was going for the jugular, but Ole Gunnar Solskjaer had an ace up his own sleeve as he brought on Fernandes.
It was Salah who had one of the best chances to win the game and complete a hat-trick, but he was denied by Dean Henderson.
Cavani still had a decisive role to play, having been at fault in the build-up to Liverpool's second goal. The Uruguayan was brought down by Fabinho - or so Craig Pawson thought, anyway - on the edge of the area.
Fernandes stepped up and pinged the ball into the bottom corner from the set piece.
Almost a year to the day since he signed for the Red Devils, the Portuguese continues to deliver the big moments.
Liverpool's horror run continues, another blow dealt to their season by their rivals - who are also six points clear of them in the Premier League table.News Now - Sport News