The Toffees, who climb to fourth place in the table, left London with a point after substitute Steve Sidwell fired a 90th-minute equaliser that secured a 2-2 draw.
It was a sickening outcome for frustrated Moyes, who watched his side's slick passing and invention create a host of chances that went begging.
"I can't believe we only got a draw out of that, it's incredible," the Everton boss said.
"We've played really well this season but that one we should have won. We were terrific at times but couldn't get the goals.
"I'm really frustrated because if I really want to hang on to the boys at the top I need to win these games when they come along.
"To win away in the Premier League is really hard for any side and we should have won that comfortably but we didn't.
"Today [Saturday] you'd have to go away saying Everton look as smart as any side. That's the way we've been playing."
Apart from Bryan Ruiz's seventh minute free-kick and Sidwell's late strike, Fulham were over-run as Everton created chance after chance.
Marouane Fellaini buried two of them, but Nikica Jelavic, Steven Pienaar and Steven Naismith were among those who failed to capitalise on the second-half fireworks.
"Give Fulham credit, they kept at it and their keeper made a couple of saves, but we were incredibly wasteful in front of goal," Moyes said.
"We were playing that well last season, the difference was Nikica was fully firing and at the end of all that good play he was the one getting us the goals. That's just form, it comes and goes.
"We went 2-1 up and Steven Naismith misses a couple of good chances to put it out of sight."
Martin Jol admitted Fulham had been taught a lesson, but offered only limited sympathy for Everton failing to claim the reward they deserved.
"It doesn't mean anything in football if you don't score the goals," Jol said.
"They dominated us, all we could do was try to be positive. They are a very good footballing team. They are a settled team and know exactly what to do.
"Normally we are a good team, but I couldn't see that today [Saturday]. Although we took the lead I still felt they were stronger.
"The only positive thing I can say is that we never gave up."