Liverpool had the lead within five minutes, thanks to Ekitike, who finished a fine ball from Ryan Gravenberch, although Hammers goalkeeper Mads Hermansen will be disappointed to have allowed it to squeeze under him.
The Reds had a second after 24 minutes, as Virgil van Dijk rose highest in the area to head home Dominik Szoboszlai's corner.
Arne Slot's side remained vulnerable, though, to West Ham's speed on the break, and some lax defending almost gave the Irons a way back into the game, particularly when Alisson Becker almost passed the ball directly to Jarrod Bowen in his own area.
A sigh of relief was subsequently breathed when Alexis Mac Allister finished brilliantly after an Ekitike knock-down, volleying into the roof of the net via an Aaron Wan-Bissaka deflection.
However, almost immediately after the restart, Tomas Soucek pulled a goal back as he ghosted in between Liverpool's defence and finished from close range. A VAR review followed but the goal was given.
The lifeline was seemingly crushed as Cody Gakpo danced his way into the box and finished with the help of a deflection, but the Hammers simply would not lie down and Taty Castellanos pulled another goal back five minutes later.
But the game was finally over on the 82nd minute; Jeremie Frimpong, on as a sub, put a cross in and Axel Disasi turned it past his own goalkeeper.
GOAL rates Liverpool's players from Anfield...
