David Moyes' side fell to a first away Premier League defeat of the new season - despite putting in a decent performance at Anfield.
But they can return to London with heads held high after a spirited display in which they created more than enough chances to get something out of the game.Tomas Soucek, back in the starting XI enjoyed West Ham's first big chance of the game when his goalbound header, from a delightful delivery by Lucas Paqueta was expertly tipped wide by Liverpool stopper Allison.
Shortly after the Czech turned provider when his centre was nodded wide by Michail Antonio, who had no excuses for missing the taget from just seven yards and with the goal at his mercy.
And West Ham soon paid for those spurned opportunities when in one of the least surprising moments of the day, Mo Salah levelled from 12 yards - having won a penalty with a trailing leg, bait that Edson Alvarez rather naively took.
The Hammers finally found their groove three minutes ahead of half time when Jarrod Bowen, back in the side having missed Thursday night's Europa League win against Backa Topola with a "virus" both began and completed a fine move that ended with him stooping low to convert a Vladimir COufal centre - the Czech's second assist in as many weeks.
Most of Anfield thought Liverpool had retaken the lead in the 45th minute when Curtis Jones turned home a deep cross, only to see the effort ruled out for offside. Yet that was simply delaying West Ham's misery - as, temporarily, did former West Ham transfer target Darwin Nunez's blaze wide from ten yards out with the goal gaping in front of him.
West Ham may have taken the lead just ahead of the hour mark when goalscorer Bowen sent his header from a Ward-Prowse corner goalbound, but one again Allison was on hand - quite literally, in this instance - to clutch a header that, directed anywhere other than directly at the 'keeper, would surely have resulted in a second goal for the #20.
And as was the case in the first half, the Hammers paid for their profligacy when the recently-embarrssed Nunez atoned for his error by lifting a chipped centre over the outstretched hand of Alphonse Areola - a goal that shared more than a little similarity to one conceded against Manchester City last weekend.
Despite being behind, West Ham were still in with a shout of taking something from the game until the 84th minute when the set-piece experts were undone from a strightforward corner that culmiated in Diego Jota adding a third for Liverpool from close range.
Having faced a pair of the Premier League's strongest teams in their last two outings, it's a slightly less daunting task facing Moyes' squad in midweek when they come up against League One Lincoln City in the Carabao Cap - before hosting Sheffield United, beaten 8-0 at home by Newcastle United today, in the Premier League.
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