Mohamed Salah scored the winner in a tense and tight match that could reignite the Reds' season.
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But Pep Guardiola was the target of coins hurled from the home fans after City had a Phil Foden goal disallowed following a VAR check.
The Spaniard was not hit by any of the missiles, but Jurgen Klopp apologised and Liverpool have vowed to ban any culprits identified for life.
It was not the only unsavoury incident on Sunday with Liverpool also expressing their disappointment at chanting and vandalism by some City fans referencing the Heysel and Hillsborough stadium tragedies.
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While there was chanting during the game, graffiti was also left in a number of areas of the Anfield Road End, which houses visiting supporters.
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Former Charlton, Crystal Palace and Newcastle midfielder Darren Ambrose expressed his dismay at the behaviour of supporters.
He told talkSPORT: "If it's the minority, so be it. But what is wrong with people? What is wrong with football supporters?
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"Going to games, throwing coins, singing vile chants about tragedies, what is going on? It's a football game, go and support your team.
"Boo your team if you want to, but this? It's pathetic and needs sorting.
"We've got the World Cup coming up, you are going back to the old ages now. It's ridiculous."
Liverpool legend Graeme Souness also reacted to the scenes on talkSPORT Breakfast, and despaired at the content of the chants heard from the away end.
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"These people are rejoicing in people losing their lives," he said.
"We saw it recently when our Queen died - people were actually celebrating that, celebrating an old lady dying, it's pathetic. These people have no place in our society.
"If you can single them out, they have to be punished. What planet do you live on when you see life like that?"
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Dietmar Hamann, meanwhile, called on clubs to take responsibility for fan behaviour as he condemned the chants heard from the City end at Anfield.
The German, who played for both Liverpool and City, told talkSPORT: "There is no place for these chants.
"I think the clubs have to take responsibility and educate the fans. It's too easy to say the FA has to act and it's something for the authorities, I think the clubs have to be proactive and try to educate these people. The FA can only do so much and it hasn't changed an awful lot in recent history, as we saw yesterday.
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"It's not the first time it's happened and unfortunately it won't be the last time. What are you going to do? You can ban them for five years, or ban them for life even, but we have to to eradicate these chants.
"There's no place for these things in football. We have other problems in the world right now, much bigger problems - this is just a rivalry between the two best teams in the Premier League over the last five years."
Dom King was present at Anfield, and the the Daily Mail's Northern Football Correspondent added on the Sports Breakfast: "Chants about Heysel, chants about Hillsborough, it just makes you recoil when you hear it.
"That song about always the victims, it is just appalling. Anyone who sings it, doesn't have a clue what this city has been through.
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"Someone threw a coin at Pep Guardiola, there were other things that were thrown. It is just ridiculous. Don't come to a football stadium if that is what you are going to do."