Everton won 1-0 thanks to Abdoulaye Doucoure's strike, with Moyes admitting Forest are a team he wants to follow next season.
He told evertontv: "Let's be fair, if you gave me next season what Nottingham Forest are doing, that would be exactly what I'd want to try to do – trying to get Everton challenging at the top, trying to get to semi-finals and cup finals.
"Our form and the way we’re going about it, we’re in that area. It's how we can take it on to next season, how we get the crowd behind us. The crowd today were unbelievable, they really were. But I think today they were looking at an Everton team that probably played as well since I've come in. They played really, really well.
"I think there has been a lot of improvements made and in recent weeks we’ve done a lot of good things. We've played against teams who we've not been able to get the ball off, and we've defended really well and stuck at it. But today I thought we were the better team from near enough the first minute.
"We had a great start to the game, which we didn't do against Arsenal, but we started the game particularly well today. We grew into it and then in the second half we played really well.
"I think it was a really good performance from the defenders, from the midfield players. And as the game went on, the attacking players grew into the game as well. And if anybody was probably going to score, it was going to be us."