The Toffees secured a point from a lifeless match at Craven Cottage but still went bottom after West Ham defeat Tottenham.
It ended a miserable week for Moyes, who saw his side dumped out of the Carling Cup by Brentford in midweek.
"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."
Everton, who must wait another week for their first Premier League win of the campaign, controlled the match but failed to make the crucial breakthrough with Yakubu missing a late sitter.
Strikers Louis Saha and Victor Anichebe are injured and Moyes lamented their absence.
"We had lots of possession and we created more chances than Fulham," he said.
"We had some opportunities later on and one or two in the first half.
"I can't come away saying we played badly though maybe we didn't create enough chances.
"We're lacking in real firepower but you'd have to say we had the best chances in the game but couldn't take them.
"There's a lack of belief in front of goal. Nothing's going in and we can't even deflect one in at the moment."
Tim Cahill started after two matches out with a knee injury and Moyes was pleased with his contribution.
"Cahill has a bad knee and he did very well to play for us today. We always knew it would be bit of a gamble," he said.
"He wanted to play and we didn't know how long we'd get out of him. To get an hour was quite good."
Fulham manager Mark Hughes, whose side remain unbeaten in the Premier League, does not expect Everton to be propping up the table for long.
"Everton showed today that they're in a false position and it's clear they're a very good side," said Hughes.
"They're at the wrong end of the table but I don't anticipate that lasting for long.
"In most of the games I've seen of them on video and TV they've played well and had impact. They've been unlucky not to get more points.
"They probably just need a win to get them going again. I'd imagine no one in the Everton camp is too concerned."