The Toffees secured a point from a lifeless match at Craven Cottage but still went bottom after West Ham defeat Tottenham.
"It's one way up now. Someone has to be bottom and it's us at this present time," said Moyes. "I don't think we look like a bottom-of-the-league team but we are so we have to get out of it. It's up to us to put it right.
"We just have to keep doing what we're doing. We'll keep playing.
"There was no lack of confidence from us against Fulham. We tried to do the right things.
"The spirit's good. The players are all together and are all working for the cause to get a result.
"My belief is that we have a really good team. That's never changed."