Bilic has been under pressure at West Ham for some time and the boss concedes talks with the chairman have been scheduled.
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.
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.
Joel Matip's first of the season also arrived from a corner, this time in more conventional fashion three minutes later, as the defender converted the rebound after Hart's save from Mark Noble's deflection.
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.
"The worldwide break does not change my situation, it makes no difference".
Jailing of Catalan politicians could lead to backlash
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Salah stayed rather too high up the pitch, with he and Firmino playing into the hands of West Ham's three-man defence.
This looked set to be the game's key tactical battle: could Salah help Liverpool break down a deep defence? This was not one of them, although the Hammers were just dreadfully flat, short of attacking spark or interplay between their attackers.
West Ham owner David Sullivan has traditionally been unwilling to sack managers mid-season, or pay compensation to appoint replacements and Moyes would be available without a cost. "It's not the first one, it's the second in a row at home".
West Ham came close to making it 3-2 in the 62nd minute from a free-kick from the left by Lanzini, but Javier Hernandez's flicked header went inches off target.
"I had the confidence to come to England again, I am not surprised", said Salah when he was asked post-match if he could have imagined his start to life in a red shirt would go so well.
"I watched him a lot at Chelsea, and it didn't really take off for him when he was a Chelsea player", he said on BT Sport.