Goals from Salah and Joel Matip inside three first-half minutes opened up a 2-0 lead for the visitors as West Ham left to boos at half time at the London Stadium.
Salah's 12th goal of the season on 75 minutes wrapped up a third win in as many games for the Reds and restored their place in the Premier League's top six.
West Ham's early solidity was undone by some very sloppy defending as Liverpool countered from a Hammers corner to open the scoring on 21 minutes.
Less than three minutes later, Matip opportunistically pounced from close range after Hart had been forced into a good reaction save when a corner was flicked on at the near post and deflected off Mark Noble.
Bilic brought on former Liverpool striker Andy Carroll at half time, but it was a cross to the more diminuitive Lanzini which gave West Ham an avenue back into the game.
The Argentine held off Joe Gomez at the back post to control Ayew's deep cross on his chest, and then finished skilfully with the outside of his foot past Mignolet.
West Ham's hopes of turning the game around lasted less than a minute as Roberto Firmino brilliantly swiveled away from Winston Reid to release Oxlade-Chamberlain for an initial shot was saved by Hart saved, only for the former Arsenal midfielder to knock home the follow-up.
Lanzini missed West Ham's best opportunity to restart their comeback almost immediately, blazing over the crossbar from eight yards after good work from Carroll.
More lacklustre defending helped Liverpool seal the three points as Mane picked out an unmarked Salah, who fired a left-footed shot across Hart and into the bottom corner for his sixth goal in five matches.
Liverpool smelled blood as West Ham threatened to capitulate in the final 10 minutes, but Firmino and substitutes James Milner and Dominic Solanke all missed chances to increase the margin of victory.
Bilic will be relieved that no more damage was done late on, but still faces an uncertain future as Hammers boss after a fourth game without victory left them 17th in the table, only a point above the relegation zone.