Idrissa Gueye struck twice as Everton defeated Southampton in their final game at Goodison Park ahead of next season's move to Bramley-Dock.
Moyes said: "Extraordinary atmosphere. The crowd were amazing, as it was outside before the game. We started the game pretty well and got in front.
"We played some good bits but not always. The job was to win the game today and see ourselves out at Goodison in a good light.
"The support at Everton is immense. It's needed some form of success. Today is a day they've been waiting on for so long, a new stadium coming."
"I'm honoured"
Moyes also stated, "I never thought I'd be the one to do it (lead the club into the new stadium). I'm really pleased. I'm honoured to be given that opportunity. We all see it as another chance. We need to start rebuilding Everton again.
"We've done the job. We have to try to build on that next year. If we can get an atmosphere like this in the new stadium it'll give us every chance of winning games.
"I thought it was as if everyone had come together as one club. Today was everyone standing together."