The result, which saw Everton midfielder Abdoulaye Doucoure, Liverpool's Curtis Jones and their manager Arne Slot sent off after the final whistle, marked the final Merseyside derby at Goodison Park before next season's move to their new stadium.
Ahead of Saturday's trip to Crystal Palace, Moyes said: "We are smiling. We were smiling with a minute to go in the game. We took a point from the game but not three. But we are also realistic that we got something from a game that was difficult to get anything at all. That made it a little bit more special and it has given us a good feeling."
On where the game ranked in his career, Moyes said: "I get asked these sort of questions quite a lot and I've been fortunate recently to play a European semi-final and final. At the moment, it's a different situation for me here. Every game you play as a manager is your biggest game. It always is because you could be fighting for you job, to stay up, a league title or European places. It's difficult to decide which game is bigger than the other.
"At the moment, it feels like a really big game. When you are a new manager, you hope to make a good start. You want get the crowd on your side and I think we have done that."
Moyes also praised the form of Beto, who now has three goals in two games, and Jack Harrison.
He said, "The people who have played as a striker will tell you they need some time to settle in. They live off their goals as well because they judge on them as well. Beto is doing that at the moment. He has come in and he's got goals. He was a difficult opponent for Liverpool and he has started very well for us. I hope he keeps it up."
On Harrison, who came on as a substitute against Liverpool, Moyes added: "I think he also played really well in the Bournemouth game. He was needing a bit of a confidence - the same with Jesper Lindstrom. Both of them are beginning to show a lot more. Hopefully, Jack is getting that confidence back like a lot of players have been."
On Doucoure's red card, Moyes also stated: "He is fine. I was disappointed he got sent off. The club will look at the situation. It was an emotional night and a game where everybody was involved. We were playing against a really good Liverpool team, probably the best in Europe, and we had to do as good a job as we could.
"I feel a bit for Arne Slot too because it is the sort of thing I was always getting involved in when I was a younger manager. It tells me that he cares a lot about his club and is fighting for his players.
"I was disappointed in Doucoure because he played so well in the game and did a great job for the team. He wouldn't have probably done what he did. But it still doesn't mean that people are allowed to abuse you. If that's the case, it's completely wrong."