Jurgen Klopp's side ran riot at the home of their greatest rivals, netting four times in the first half to leave Man United facing their biggest ever scoreline at the halfway stage of a Premier League match.
Mohamed Salah, who had scored two of those first-half goals, then completed his hat-trick early in the second half to make it 5-0 to Liverpool.
Man United thought that they had pulled a goal back when Cristiano Ronaldo tucked home from inside the box, but the goal was ruled out by VAR for offside.
Half-time substitute Paul Pogba was then sent off for the hosts to add insult to injury, but Liverpool were content with their five-goal advantage and did not push particularly hard to add to their lead despite facing 10 men.
The 5-0 scoreline is the heaviest Man United have ever suffered at the hands of Liverpool at Old Trafford, while it is also the first time they have lost to nil by five or more goals at home since February 1955 against Manchester City.