The 56-year-old Moyes saved West Ham from relegation in 2017-18 and was named Manuel Pellegrini's successor on Sunday after the team fell to 17th in the table following a 2-1 home defeat by second-placed Leicester City.
"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," Noble told West Ham's website.
"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."
West Ham, who have lost seven of their last nine matches to sit one point above the drop zone, host 16th-placed Bournemouth in Moyes's first match in charge on Wednesday.
"My main thing is that I have always put the club before myself in my career. I just want us to do well," Noble added.
"We hoped that would happen with Manuel because he is such a great guy and had a proven managerial pedigree, but we now need to get back to what we are as West Ham and achieve realistic targets before aiming higher.
"I know it's a tough time for the fans... but I do believe we need to come together and show the manager support and start getting results because... results are all that matters."