The genial Croatian has been under pressure at West Ham pretty much since they left Upton Park.
There have been occasions when West Ham have been genuinely awful under Bilic.
The in-form Andre Ayew came into this clash with three goals in his past two outings and he nearly added to that tally in the ninth minute.
"Our counter-attacking was pretty flawless, we had some wonderful play", Liverpool manager Juergen Klopp, who was boosted by the return of Sadio Mane after a month out, said. Salah's scruffy low corner hit the hapless Mark Noble, forcing Hart into a decent save, and Matip couldn't miss with the rebound.
However, Firmino's brilliance allowed Liverpool to quickly double their advantage again, just 57 seconds after Mignolet was picking the ball out of his net.
The result left them precariously perched above the relegation zone and sent Bilic into the worldwide break with question marks hanging over his future.
He turned Winston Reid inside-out on the halfway line, leaving the Kiwi mesmerised, and fed Alex Oxlade-Chamberlain who got his first Premier League goal for the Reds at the second time of asking, with Joe Hart saving his initial effort. Sane picked the ball up outside his own penalty area, raced into the Hammers half, stormed in on goal then unselfishly passed to the in-coming Salah.
Matteo says that West Ham should give Carroll an hour and see if he can cause problems for the Reds, dismissing the view that the England striker is better being brought on as an impact substitute. "It's not the first one, it's the second in a row at home".
Asked after Saturday's defeat if he still had West Ham's backing, Bilic told BT Sport: "It's hard for me to talk about that now".
Huddersfield held on for a 1-0 win over West Bromwich Albion despite playing the final third of the game with 10 men.
"I have to talk to the chairman, we are going to discuss this defeat", he said.