Everton came from behind to win thanks to Bernd Leno's own goal on 83 minutes.
Moyes said afterwards: "I thought we done quite well. I didn't like parts of the first half. I didn't think we got near their goal too often, and we gave away a bit of a poor goal. Not a poor goal – the keeper actually makes a great save but then it goes on to Myko for an own goal.
"But we gave them one or two opportunities after actually starting quite well. I thought we passed the ball quite well but the bigger problem was I didn't think we went anywhere, we kept it around the halfway line and didn't really make a dent on their defensive line. In the second half, we certainly did a much better job."
On securing a rare win at Craven Cottage, Moyes was particularly pleased.
He also said, "Well, look, for us to come here and win at Fulham is an incredible result. I mean, we've just come away from Brighton with a draw. We've now come to Fulham and picked up three points. Both teams, whether it be Brighton or Fulham, are very close in the league table, so for us to keep picking up points is so important.
"We make sure that we're definitely not going to be in amongst the bottom teams now. The challenge really for us all is to try and make sure that we can try and challenge whatever's above us. Whether we can or not, whether we're good enough, whether we can get the players back to the levels they were probably in September-October, that's something we need to try to do if we can.
"But it was a great performance today from the players. They did a brilliant job, they really did. We turned it around at half-time. They worked, they were more intense, they were quicker, faster, we played well, our football was good."