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."
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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.
"The clubs have to educate the fans. It is too easy to say the FA have to act. The clubs have to be proactive and educate people.
"It is the not the first time this has happened. There is no place for these things in football."