The Toffees kicked off their Premier League campaign with a comfortable 2-0 win over Crystal Palace, thanks to goals from Kiernan Dewsbury-Hall and Thierno Barry.
Everton have now beaten Championship Preston 4-0 to reach the third round of the EFL Cup, with Barry staring once again, scoring a hattrick.
Things are looking positive for Everton at the moment, but Moyes has urged the fans not to get too carried away.
"It ended up a bit more comfortable than I thought it might’ve been but obviously a really good result,” he told Sky Sports.
“We played well at times, our football was quite good. We needed to finish things off a bit better in the first half but really pleased we scored four goals."
On whether fans can get carried away: "No they can’t. I think we’ve got a lot of work to do and I still feel that for us to be a side which is competitive I still feel that we’ve got a lot of work to do. Hopefully we’re able to try and do some of it.
"More players. Our squad’s not big enough. We need to get more quality in some areas if we can do."