Michael Keane fired Everton ahead with an early header before Jarrod Bowen struck an equaliser in Nuno's first game in charge of the Hammers.
Moyes later said: "We played some decent stuff at times, we got a goal but we couldn't get a second one which would have been all important. We did a lot of good things but we just couldn't quite craft a chance to get a second goal.
"Once we didn't get it, West Ham grew in confidence and belief that they could get something themselves which they did. Some good things and some not so good things but West Ham have talented players so it was always going to be a tough game for us."
On needing to adjust his tactics given Nuno's appointment, Moyes continued: "I don't know about a different West Ham, I just know the capability of the players obviously I have had them before and worked out how good they are.
"We are talking about Brazilian internationals, England internationals so anybody thinking that they haven't got a good team is wrong. They have a very good team and at times they showed it tonight. We played well tonight but we just couldn't find a clear chance to get a second goal."
Asked if the players are now feeling more settled at their new stadium, Moyes added: "It's certainly one we are more familiar with but we want to win. Our record here at the moment has been good enough, we want to keep it going. We started the season well. We want to raise expectations but we can't forget that we have a long journey to go and we are just at the start of it."