The Blues had several chances, with Beto twice denied by Mark Flekken in one-on-one situations during the first half.
Their frustration grew when Yoane Wissa capitalized on a defensive lapse, but Everton battled back to extend their unbeaten run to seven games.
Post-game, Moyes said: "I think we deserved more in the end, I really do. I think our point was probably the least we deserved.
"We lost a terrible scrappy goal right on half-time from a long throw which, everybody who comes here is well aware of what's going to happen, and we didn't deal with it well.
“But look, I thought we were unlucky. Beto had been through one v one which he’s recently done very, very well on. Then he had the second one as well. If we hadn’t got a draw there would have been something wrong.
"(The unbeaten run) is good because when you come in as manager you want to get on a run of games we needed to make sure we’re going to stay in the Premier League. I think tonight’s another point towards it.
"On the road we've just won at Brighton, we've won at Crystal Palace. In truth, we should have probably won here tonight as well.
“So it says a lot for the players, what they're doing. Big Jake gets his first goal tonight which was great for us as well. So there’s lots of good things from the team.
"I thought they played well, really well in the second half. You mustn't forget that we're still without probably the best part of eight or nine of our senior players."