The Hammers have taken just one point from four matches under the guidance of the Scottish tactician but left the Etihad Stadium last weekend with their pride intact having run league leaders Manchester City close.
Last Sunday's 2-1 loss leaves the east Londoners 19th in the Premier League heading in to the busy festive period but Noble says the spirit within the dressing room remains high given the visible signs of improvement under Moyes.
"I'm not going to stand here and say it's been fantastic because, for me, football is not enjoyable unless you're winning games," Noble told Sky Sports News.
"What I will say is that it's been good under the new manager, he's set a stall out for how he wants to play. Some of the performances have been really good but the results haven't gone the way we wanted. You can slowly see it turning around. If you don't stick together you'll be in trouble."
The West Ham skipper has now turned his sights towards Saturday's game against Chelsea, a match that comes amid a tough run of fixtures for the London Stadium outfit, and Noble believes they can claim the scalp of the reigning champions.
"We're going to have to pull a couple of big guns out the hat," said Noble. "Chelsea are a fantastic team, I've watched a lot of them, they've got great players.
"But we played really well against Manchester City last weekend and if we play like that we'll have a chance."