The Reds, who recorded a whopping 36 shots during the match, took the lead through Sadio Mane but were then pegged back nine minutes later as Jesse Lingard pounced on Alisson's error.
However, summer signing Xherdan Shaqiri came off the bench and scored twice to seal Liverpool's fifth consecutive victory in all competitions.
With the score somehow level at half-time, United manager Jose Mourinho tried to change things by bringing on Marouane Fellaini for the yellow-carded teenager Diogo Dalot.
In truth, the big Belgian was no worse than Ander Herrera or Nemanja Matic, but there was one truly embarrassing moment which led to him injuring a steward.
How Fellaini's arrival changed Man Utd's setup
United started the game with a back three of Matteo Darmian, Eric Bailly and Victor Lindelof, while Dalot and Ashley Young played as wing-backs.
When Fellaini was introduced at the interval, Mourinho's men reverted to a four-man defence and the versatile midfielder was given a bit more license to try to get forward.
For all his limitations, there aren't many footballers who can bring a ball down using their chest as well as Fellaini, and he showcased that skill again during the 3-1 defeat.
Midway through the second period, the 31-year-old got in between Nathaniel Clyne and Jordan Henderson and nicely controlled the ball into Liverpool's penalty area.
Fellaini manages to injure Anfield steward
However, as you can see in the videos below, not only did the shot Fellaini unleashed go nowhere near Alisson's goal, it also managed to knock an Anfield official off their chair.
Whether the person was male or female, a steward or a paramedic, they were clearly caught completely unaware by Fellaini's strike and amusingly couldn't maintain their balance.
What next for Liverpool and United?
After seeing off Napoli to reach the Champions League knockout round and beating bitter rivals United, Jurgen Klopp's men face Wolves and Newcastle United next.
Man Utd, now 11 points behind fourth-placed Chelsea, are desperate for victories from games against Cardiff City and Huddersfield Town as they attempt to ressurrect their 2018-19 campaign.
What do YOU make of Fellaini's shot knocking an Anfield spectator off their chair? How far behind Liverpool do YOU think Man Utd are? Drop YOUR thoughts in the comments box below!