Moyes, leaving for Manchester United at the end of the season, watched his Everton side beat West Ham 2-0 in his last home game in charge thanks to two goals from Kevin Mirallas.
The outgoing manager reserved his most fervent praise for the home support, who gave Moyes a fantastic reception before, during and after the game.
"I thought we played as well as we have in a long time," Moyes said.
"We scored a well-worked goal in the first half and but for a couple of great saves and a few bad misses we would have had more.
"The crowd showed today what a big club Everton is, and I think today that was their part in it all. It was an emotional time during the second half because they got right behind their football club. They got the team that second goal.
"I think the crowd today showed what Everton means to them.
"I count myself extremely lucky. I have had a really good chairman and board who will let me get on with things. That is what Everton do: they want stability."
The result confirmed Everton's top-six finish in the Premier League, whilst West Ham are yet to ensure a top-half place.