After Leandro Trossard's opener for Arsenal, Everton fought back through a successful penalty scored by Iliman Ndiaye, won by Jack Harrison.
Afterwards, Moyes said: "Really satisfied. It has been a difficult week - the team who are top of the league and the team that are second in the league. We gave them a good game.
"The job is nearly done. Mathematically, we are not safe yet. I am hoping to take a visit to the new stadium this week, which would indicate that I think we are close. We need to get used to playing Premier League football there."
On Ndiaye, Moyes continued: "We have missed him. We have not had Dwight (McNeil), we have not Ili since the Liverpool game at home, Dom (Calvert-Lewin) and (Armando) Broja, too. The players who have been in, have done very well.
"Off the ball, we have had two games where we have had to do most of the work out of possession."
Everton now face top six teams in their next three games.
"I am actually looking forward to the games to see how we compete. In the two games (against Liverpool and Arsenal)
we have showed resilience and we stuck at it.
"Even today when we did not play particularly well in the first half, we made one or two opportunities. I am looking forward to playing the games because I am interested to see all the players and how the play."