The Toffees boss was left frustrated after Steve Sidwell scored at the death to cancel out Marouane Fellaini's two goals and earn Martin Jol's team a share of the points
A Tim Howard own goal had sent Fulham into the lead in the seventh minute before Fellaini gave the Toffees control, but Sidwell earned Martin Jol's team a point.
Moyes rued his side's missed chances and admitted that his side's lack of an in-form goalscorer was a problem, with Nikica Jelavic enduring a mixed spell.
"I can't believe that we've only got a draw out of that, but give Fulham credit, they kept at it," Moyes told reporters.
"We were incredibly wasteful in front of goal, but the way we played, the chances, the opportunities, when we played that well last year the difference was we had Jelavic finishing off the chances.
"At the end of our good play he was the one getting us the goals and I think he's just a little bit off-colour in terms of his goals at the moment. We've had quite a few games like that this season where we should have been away.
"We had three shots on target in the first five minutes before they come up the other end and score directly from a free kick. We found ourselves a goal down and that was hard to take and we found ourselves chasing the game."
Moyes admitted that Everton needed to turn their chances into goals if they wished to remain among the chasing pack at the top of the Premier League.
"You can see the quality we've got," he insisted. "We can be terrific at times but today we just couldn't get the goals and it's frustrating because we're trying to hold on to the boys at the top and we need to try and win the games when they come along.
"Winning away in the Premier League is very hard, against any side, and we should have won this one quite comfortably and we didn't."