Fulham had the lead through Raul Jimenez, but Everton cruised to a 3-1 win thanks to goals from Vitalii Mykolenko, Michael Keane and Beto.
A pleased Moyes told evertontv afterwards, “I was worried about coming here today because they're going for Europe, and I'm thinking, my goodness, they probably know that it's one of their last home games.
“We didn't play well for 30 minutes or so, then got ourselves back into it. The only thing that broke for us the whole day was Myko’s shot and deflection, but we went on to play really well in the second half and were worthy winners in the end.”
Moyes was also happy for Keane, who stepped in for an injured Jake O'Brien.
The manager continued: “He’s a good guy. He's probably had a few tough times recently, and it's not been easy because of the form of Jared and Tarky, mainly and even Big Jake. But I thought he came in and he played so well today.
“He scores a goal from a set-piece which we've found really hard to do since I've been here.”