Moyes appears to be instigating a revival at his new club, with Saturday’s 3-0 victory at Stoke their third consecutive game without defeat.
Before then, Moyes’ side had lost 2-1 at Manchester City before beating Chelsea and drawing against Arsenal at home.
They have climbed out of the relegation places and now sit 15th in the table ahead of tonight’s Carabao Cup quarter-final away at Arsenal.
Like West Ham, Moyes admits he is also rebuilding his reputation after ill-fated spells in charge of Manchester United, Real Sociedad and Sunderland.
“I have to come here and show I can do it,” said the Scot. “If you’re any player you have to come and show what you can do. Your reputation doesn’t stand for anything.
“You have to come here and try to get up and show you’re capable of doing the job. I think I’m capable of doing the job at any club in the world so I’m sure I can do it at West Ham.”
One player to have enjoyed a particular upturn in form is Marko Arnautovic, who cost West Ham a record initial fee of £20million in the summer and has scored twice in his last three games.
Arnautovic provoked anger at Stoke by celebrating in front of his former club’s fans, as well as slapping the badge on his shirt and crossed his wrists - the West Ham salute.
Stoke’s supporters hurled abuse at Arnautovic throughout the game and Mark Hughes exchanged words with the forward when he was substituted in the second half.
“Of course he should be celebrating, I think actually people want to see the celebrations, don’t you?” Moyes said.
“All he did was slide to his knees, I don’t know how that in any way was a bad way to have done it.”
Arnautovic may be among those rested against Arsenal tonight given Moyes is likely to have one eye on Saturday’s crunch Premier League game at home to fellow strugglers Newcastle.
Joe Hart will start the match in place of Adrian in goal while Andy Carroll could play after shaking off a back complaint.
Mark Noble, however, will sit out after Moyes said the midfielder had picked up a problem with his hamstring.
Arsene Wenger will rotate his squad for the Hammers clash and has backed the latest crop of young players to become the future of the club.
The Frenchman has a history of using the competition as a way of blooding talent in the first team and that has been the case again this season.
With his side having looked lethargic in Saturday’s 1-0 Premier League win over Newcastle and on the back of two away draws against Southampton and the Hammers, it is clear that Wenger will once again shuffle his pack.
While that could mean starts for international players such as Danny Welbeck, Mathieu Debuchy and Mohamed Elneny, it is also a chance for Wenger’s young ensemble to impress.
Ainsley Maitland-Niles has started the last two league games having caught the eye in the Carabao Cup and Europa League this campaign.
And Wenger will continue to give chances to his young stars, such as Matt Macey, Joe Willock and Eddie Nketiah – who scored twice off the bench against Norwich in round four.
“It’s a very exciting bunch of young players,” Wenger said.
“I made the decision a few months ago to try and keep them at the club because I believe they all have futures at the club.
“For example on Saturday, we had four players who have been developed in our academy – Alex Iwobi, Jack Wilshere, Hector Bellerin and Ainsley. At our level you have not many teams who can show up with that.
Macey, Nketiah and Willock are just three of a number of youngsters to sign new deals with Arsenal recently and – if they do start against West Ham – are likely to line-up alongside players on the fringes of Wenger’s first team.