At one stage on Sunday, the Toffees were 3-0 ahead before Spurs' late rally.
Moyes said afterwards: "When you're in the position we're in we know any win is a good win. In the first half I thought we played really well and did a lot of really good things and maybe could have scored a couple more.
"We did very well and put ourselves in a good position. The thing was to go and get the fourth goal but then it was a nervy ending. I knew if they got one it would be difficult because of how they play but thankfully we saw it out.
"The players have probably needed a bit of confidence and belief. We're thrilled to get three points. It was a difficult game but we did such a good job in the first half.
"Dominic (Calvert-Lewin) got one and I'm sort of banging the drum that we need him to get more because we need goals. I thought there were other opportunities and there is a long way to go but it is a good start. We need goals from the front players and if we can get Iliman Ndiaye closer to the goal then we think he's a threat.
"We've got a lot of experienced players here and they'll be disappointed with themselves for the game management and how we gave the ball away in the second half. The game management wasn't the best and we need to work how to do that better.
"I'm only getting to know the players but I don't think this changes our thoughts on the transfer window. We need to bring a couple of players in. We need to make sure we're playing in the Premier League in the new stadium. We do need a couple of additions."