There was little sympathy for the beleaguered Bilic from Liverpool, who wrapped up a resounding success with Salah's 12th strike of the campaign, lifting Jurgen Klopp's men back above Burnley into sixth.
West Ham had actually started okay and Lanzini struck the post when it was 0-0.
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.
Defender Steve Cook headed in the 92nd-minute victor for Eddie Howe's strugglers, earning them a second win from their last three Premier League games.
Despite his unsuccessful time at United and further unproductive spells at Real Sociedad and Sunderland, he boasts a wealth of managerial experience and is seen as someone who can steady the ship at West Ham, the source said.
But it didn't have the desired effects, and the Baggies came away with nothing on an afternoon where Gareth Barry played against his 46th different team in the Premier League (bear in mind there have only ever been 49).
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This looked set to be the game's key tactical battle: could Salah help Liverpool break down a deep defence?
A 4-1 thrashing at the hands of Liverpool seems to have been the final straw for the club and Bilic, with the Hammers sitting just above the relegation zone heading into the worldwide break.
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However, speaking to BT Sport, Klopp explained that the Dutchman had a good night and the swelling had "disappeared" by the time the players convened this morning.
With 12 goals in 17 games, Salah has certainly settled into life at Liverpool; factor in his four assists and his contribution to Klopp's side can not be underestimated.
"I said yesterday he was ready for 25 minutes but it was 25 minutes plus and it was good for us".