In an open letter to fans, midfielder Noble, 32, said Moyes was "really liked and respected" by Hammers players who had worked with him during his first spell in charge at the club.
Noble declared: "As captain of this great football club, I want to speak directly to you, my fellow West Ham United supporters.
"I know that we were going through a difficult time and the club decided to make a decision to change manager.
"David Moyes knows the club and he knows a lot of the players and the players he worked with - myself, Manu, Angelo, Arthur, Cress, Dec, Reidy, Michail and Zaba - really liked and respected him and the way he worked when he was here. He got players playing to their full potential and I hope and believe he can do it again.
"I know the fans are not happy with the situation we find ourselves in, but we all love and support our club and we need to get behind the manager and try to get ourselves out of this position.
"The manager worked for a long time at a club that is very similar to West Ham in Everton, and you don't stay at a club for that long under the scrutiny of passionate fans unless you are good at your job.
"I think David is the best person for the job."