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.
It could have all been so different, though, had West Ham taken the glorious chance which came their way after nine minutes.
Lanzini’s ball over the top, with the aid of a deflection, fell at the feet of Andre Ayew.
With Simon Mignolet a little slow to come out, Ayew lifted the ball over the goalkeeper only to see his effort hit the outside of the near post.
But in the 22nd minute it was the Reds who went ahead – straight from a West Ham corner.
After Mark Noble’s cross was headed clear Salah clipped the ball to Sadio Mane and the duo charged upfield.
Mane was a surprise inclusion in the starting line-up having made a quicker than expected recovery from a hamstring problem.
The Senegal striker was certainly showing no ill-effects from the injury as he hared the length of the field unchallenged.
Mane reached the edge of the West Ham penalty and, with only Aaron Cresswell for company, squared for Salah to roll the ball past Joe Hart.
It was a devastating counter-attack and left the hosts shell-shocked after their bright start.
But things got worse two and a half minutes later when Salah’s corner hit Noble, forcing Hart to make a save.
The England keeper could only parry the ball straight to Matip, who buried the rebound before a West Ham defender could react.
Bilic, now relying on the sort of second-half display which saw his side come from two down to beat Spurs in the Carabao Cup last week, turned to Andy Carroll for inspiration.
And when Lanzini held off a weak challenge from Joe Gomez at the far post before tucking in Ayew’s cross, the fightback looked as though it might be on.
That hope lasted all of 55 seconds, though, as Liverpool merely sauntered up the other end with Roberto Firmino teeing up Oxlade-Chamberlain for his first Premier League goal for the Reds.
Salah wrapped up the scoring – and sparked another exodus of angry punters at the London Stadium- when he collected Mane’s pass on the corner of the area and expertly placed his shot past Hart for number four.