Having dispensed with Pellegrini’s services following Saturday’s soul-sapping home defeat by Leicester, West Ham wasted little time in turning to the man who saved them from relegation two seasons ago.
It was not a moment too soon, with the Hammers kicking off in the Premier League drop zone following Aston Villa’s win at lunchtime.
Nine defeats from 12 matches, including the last four at home, had left morale in east London at rock bottom.
But in just two training sessions Moyes seemed to have instilled more motivation and organisation than the team had mustered over the previous three months.
Moyes received a lukewarm reception from the home fans before kick-off, but those underwhelmed by his appointment were not complaining by the end.
West Ham took a 17th-minute lead when Robert Snodgrass exchanged passes with Anderson and reached the byline before pulling the ball back for Noble on the edge of the area.
Nine minutes later West Ham doubled their lead when Ryan Fredericks overlapped down the right and crossed towards Haller.
The French striker, who had looked drained of confidence under Pellegrini, showed a glimpse of why he cost £45million in the summer with an acrobatic volley past Ramsdale.
The Hammers were in uncharted territory by the 35th minute when Noble scurried into the area and tumbled to the floor under Harry Wilson’s challenge.
Noble appeared to have tripped himself over but a VAR check deemed he had still been fouled, and the Hammers skipper tucked away the penalty.
Noble had his sights on a hat-trick after the break when Haller broke down the right and crossed towards the unmarked midfielder, but just as he shaped to volley home Cook nicked the ball away.
Instead it was Anderson who grabbed the fourth, collecting Declan Rice’s pass and outstripping Simon Francis before rolling the ball under Ramsdale.
Everything was going West Ham’s way, so much so that Aaron Cresswell even had a red card for a foul on Ryan Fraser downgraded to a yellow by VAR.
Then Dominic Solanke’s glancing header came back off the inside of a post only to bounce into the hands of Fabianski.
While the Hammers levered themselves back out of the bottom three, Bournemouth slid into it and, with eight defeats from their last 10 matches, they looked every bit a side destined to stay there.