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Both sides could claim to have been deserving winners in a classic free-flowing encounter full of goalscoring chances, remarkable saves and four disallowed goals.
Minamino came off the bench to score in a 3-0 win for Jurgen Klopp's side.
Victory for high-flying West Ham lifted them above Juergen Klopp's side into third place in the table.
With four wins out of four, Juergen Klopp's side can now take their foot off the gas for the final two games of the group stage.
Kevin De Bruyne scored in the 81st minute to make it 2-2 and give City a point at Anfield.
Sadio Mane, Mo Salah and Naby Keita scored for the Reds in the Premier League match at Anfield.
An 83rd minute penalty from Ashley Barnes, after he was brought down by goalkeeper Alisson Becker, decided the contest after Juergen Klopp's side failed to score for the fourth straight league match.
The Liverpool talisman went down under a challenge from Ramos inside the first half an hour of the Champions League final and suffered shoulder ligament damage which forced him off the pitch.
In his letter to Klopp, Daragh Curley asked the 52-year-old German coach to engineer a few Liverpool defeats so they would not win the Premier League this season.
Trailing 2-1 late in the second half, West Ham keeper Lukasz Fabianski was twice at fault as Mohamed Salah and Sadio Mane ensured Liverpool prevailed to equal the record of 18 consecutive top-flight wins set by Manchester City in 2017.
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