Quickfire first-half goals from Mohamed Salah and Joel Matip saw the hapless Hammers booed off at the interval.
Manuel Lanzini pulled one back in the second half but less than a minute later Alex Oxlade-Chamberlain extinguished their hopes of a comeback before Salah hit an exquisite fourth.
And while there remains no real appetite to sack Bilic among the West Ham hierarchy, it is hard to see how he can continually stagger on to the next 'must-win' game to save his job.
The Croatian has just survived two of those, against Tottenham and last weekend's last-gasp draw at Crystal Palace, and this loss sharpens the focus once more.
For Liverpool it was certainly a happier trip to the capital than their last one, the 4-1 Wembley defeat by Tottenham a fortnight ago.
Check out how each player rated in the gallery above.
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