The Hammers should have leveled from Andre Aywe, but he twice struck the base of the post from point-blank range in an incredible passage of play following a corner in the last action of the first half.
Liverpool eventually doubled their advantage on 57 minutes courtesy of Coutinho, who found the bottom corner with a precise finish after jinking his way into space.
The playmaker then made it 3-0 when he capped off a speedy counter-attack by firing beyond Adrian after the visitors broke at pace from a West Ham corner.
Divock Origi rounded off the scoring by steering in a fourth goal from a loose ball inside the area to put the seal on an excellent day’s work for Jürgen Klopp’s men.
Thumbnail photo via LiverpoolFC.com