Noble scored his second 10 minutes later from the penalty spot after being brought down by Harry Wilson, before Felipe Anderson completed the rout.
West Ham were lucky not to end the match with 10 men after Aaron Cresswell was sent off for a reckless challenge on Ryan Fraser, only for the video assistant referee (VAR) to downgrade the dismissal to a yellow card.
Victory was West Ham’s second in six matches, while the defeat sees Bournemouth slip into the relegation places in 18th.
Moyes replaced the sacked Manuel Pellegrini on Sunday, returning to the club he led to Premier League safety at the end of the 2017-18 season.
In his first spell at the Hammers’ helm, Moyes lost his opening match in charge – away to Watford – and had to wait four games before finally getting a win on the board.
But it was the polar opposite second time around, a classy performance from the Hammers securing a statement win on home soil.
Moyes stamped his return by making four changes to the side which had lost to Leicester on 28 December under Pellegrini, and those changes proved pivotal with Noble and Robert Snodgrass the stars of the show from the off.
Noble had called for West Ham fans to get behind Moyes after his reappointment and the captain led from the front with his opening goal, wrong-footing Aaron Ramsdale with his deflected strike after brilliant play from Snodgrass.
The scorer geed up the crowd as the match got back under way, signalling to them to ramp up the volume, and they did exactly that when Haller found the back of the net seven minutes later, through an exquisite mid-air finish from Ryan Fredericks’ cross.
It marked the first time the Hammers had scored twice in a first half this season and Bournemouth had no answer as their hosts repeatedly thundered forwards, with Noble soon scoring his second after being tripped in the box by Wilson, sending Ramsdale the wrong way with his spot-kick.
West Ham did not quite have the same control of the second half as they did the first as the Cherries looked to claw something back, but if the three points had not already been sealed by now, Anderson’s second-half strike proved the final nail in the coffin.
The Brazilian effortlessly brought down Declan Rice’s weighted pass before a simple finish condemned Bournemouth to their 11th defeat of the season.
