Lanzini, in tandem with Marko Arnautovic, caused Huddersfield’s defence constant trouble throughout the game with the Hammers looking a totally different team from the demoralised outfit that Moyes inherited when he took over from Slaven Bilic in November.
Neither the Argentine Lanzini nor Austrian international Arnautovic are out-and-out strikers but with Andy Carroll injured, Moyes paired them up-front and they played like they had been a partnership for years.
Arnautovic's strength, skill and pace showed the talent that prompted West Ham to sign him from Stoke in July for a club record fee of around 25 million pounds.
The win was the 200th in the Premier League for Moyes, making him just the fourth to reach that milestone following Alex Ferguson, Arsene Wenger and Harry Redknapp.
West Ham are now five games unbeaten in all competitions.
The Hammers took the lead in the 25th minute, taking advantage of sloppy play when after a pass from keeper Jonas Lossl, Town's Joe Lolley was robbed near the edge of his area and Mark Noble burst goalwards and fired home.
Lolley made amends in great style five minutes before the break, beating West Ham keeper Adrian with a fine curling left-foot shot to send the teams in level at the interval.
That suggested a tight battle in the 45 minutes to follow but West Ham came out and put the game beyond Huddersfield within 15 minutes.
Just seconds after the break, Arnautovic restored West Ham's lead, showcasing his technique as he hooked the ball past Tommy Smith and then drilled the ball home.
Arnautovic then sent Lanzini through with a smart through ball with the outside of his foot and the Argentine slotted home confidently to make it 3-1 in the 56th.
The second half blitz was completed five minutes later when Arnautovic broke from the halfway line and found Lanzini who blasted past Lossl from the centre of the area.
The win moved West Ham above Huddersfield and up to 11th in the table and Moyes was delighted with the continued improvement from his team.
"We have a long way to go but the players have done brilliantly well since we came in and they did very well today," he said.
"I'm really pleased with Marko Arnautovic. Everybody knew how good he was at Stoke City and maybe giving him a freer role up front has given him a chance to express himself."
(Reporting by Simon Evans; Editing by Ken Ferris and Pritha Sarkar)
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