The Reds went ahead through a brilliant goal by Sadio Mane and it was no less than they deserved after some early dominance.
But Alisson's error let Man United back into proceedings and the second half was a much closer affair, with the visitors providing their fair share of scares while Liverpool attempted to turn the screw at the other end.
Xherdan Shaqiri's introduction proved ultimately crucial with the Swiss winger coming on and scoring twice to settle this game in the hosts' favour.
But what did we learn? Here are 5 things
1. Jose Mourinho still can't make United defend properly
For a coach whose career reputation has been built on making teams hard to beat first and foremost, this iteration of Manchester United only fuels the suspicion that the Portuguese's best days are behind him.
Mourinho rails publicly and privately against his central defenders but he has had several transfer windows to resolve that. Victor Lindelof and Eric Bailly, the starting centre-backs today, were both bought during Mourinho's reign. There is no sporting director to blame. Yes, his defenders can't pass the ball but similarly he can't pass the buck in this instance.
Liverpool cut through United at will, especially in the opening stages as the Reds went in front through Sadio Mane and then looked set to pull away.
Jesse Lingard's goal knocked the hosts out of their stride a little but they were still the better side and came on strong in the second half.
Eventually the territorial advantage that Liverpool had proved crucial with two heavily-deflected strikes finding their way home and settling the game.
"Don't sack Mourinho" sang the Anfield crowd, another embarrassment for the United boss.
2. Alisson needs consistency more than hype
It's part of playing for Liverpool that a couple of good performances will immediately have you catapulted into the elite, but Alisson's up-and-down displays have been a background issues for Liverpool - coming to the fore in dramatic fashion against Manchester United at Anfield.
The Brazilian's pitiful effort handed Jesse Lingard a simple equaliser and let United back in the game at a time when they were being overrun.
It took the wind out of Liverpool's sails and the half ended far more even than it had appeared before the visitors were gifted a leveller.
Only Bernd Leno and Asmir Begovic have given away more Premier League goals with errors this season than Alisson. That's not what Liverpool thought they were getting for their huge spend.
3. Gary Neville's fears were ill-founded
Who could have guessed that launching a clothing line a week before a game doesn't actually affect your footballing ability?
On Sky Sports before this monster clash we saw Gary Neville joining Roy Keane's criticism of Jesse Lingard after he debuted his new collection for fashion label JLingz.
Lingard did excellently to haul a poor United side back into the game, making the most of Alisson's slip and adjusting in a milisecond to finish well.
Scientists are still searching for conclusive proof but the latest evidence supports the idea that promoting a textiles business actually has little material affect on footballing ability.
4. Pogba and Mourinho's relationship continues to stink the place out
Paul Pogba was left on the bench for this one.
Man United's club-record signing being a substitute for games that will define the team's season seems a little peculiar, and yet we have become so accustomed to this nonsense between a manager who wants to assert his authority and a player who dreams of being at the orders of a modern coach that we're left in a stalemate.
The result is that nobody is surprised to see Pogba benched for a massive game and even as he stays wrapped in a coat as Liverpool score a second and third there isn't even a twitch.
United look lost and are drifting into mid-table. Pogba can only watch from the sidelines.
5. Shaqiri introduction seals Liverpool win
Xherdan Shaqiri's substitute appearance could barely have gone better.
Some Liverpool fans had wondered if the in-form Swiss attacker could be squeezed into the line-up but he had to make do with a place on the bench.
When he did come on, even for a brief time, he made the most of it. Two goals in the dying stages buried Man United and handed Liverpool all three points. It was a game-changing move from Klopp and prompts soul-searching moves for United.
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