Loris Karius enjoyed another solid performance between the sticks, while the midfield was controlled by unlikely heroes for the Reds.
They are now up to 11th, four points clear of 18th place, which is occupied by their beaten opponents Swansea.
Could Real Madrid do with a lethal left-footed player after a certain other leaves this summer?
The Mexican worldwide has been in a rich vein of form in front of goal of recent, scoring thrice and assisting another in his last 4 league appearances for West Ham.
Close to 3,000 West Ham fans made the journey from east London to Liverpool and made quite an impression as they created a pocket of the stadium claret and blue. Of the 103 goals scored by the Reds, Mohamed Salah has 31 after netting the second at Anfield.
And Moyes admits they were up against a risky Liverpool side that caused them "big problems" throughout.
The desire to impress in such a high-profile encounter and also get one over on the Liverpool fans will very likely spur Chicharito into giving his best.
Jurgen Klopp's side are unbeaten in the league an Anfield this season but they have drawn six top flight matches at home. Despite the eventual scoreline, the West Ham keeper Adrian did plenty to try, unsuccessfully, to keep his side competitive throughout the game.
Nathaniel Clyne (muscular) is the only player preventing Liverpool having a clean bill of health which should provide Klopp with plenty of options for this fixture.
Having endured a awful start to life in East London, Marko Arnautovic has been a player dramatically transformed under David Moyes' guidance. Their last 10 league matches have produced 20 goals and a +6 goal difference which is far superior to the teams now around them.
The Hammers, whose supporters wore T-shirts commemorating the 25th anniversary of Bobby Moore's death, offered little and the result prolonged their manager David Moyes' miserable record at Anfield.
"I was due to be with the rugby team on Tuesday but unfortunately I couldn't get there, but Eddie's going to come and watch us train in the next couple of weeks", he said.
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