He acknowledged the intensity of the Merseyside derby and felt his side deserved something from the game.
Moyes emphasized the importance of building on this result as Everton push for a strong finish to the season.
Moyes told reporters post-game: “We're trying to improve what we can. We're trying to make things better. Tonight was a night where we had to suit the stadium, a lot of things in the game. We went more direct tonight but it suited us.
"We got it in the end and the point in the end more importantly is a point towards making sure we're a Premier League club."
On whether the point removes relegation fears: "I wish it did. I think that us picking up the points regularly is what we've got to try and do between now and the end of the season. I think there's a group here who lack some quality in areas but we've got to try and fill in, we've got a few injuries but hopefully there are signs of one or two coming back."
On Everton scoring more often lately: "This has never been an easy ground for anybody to come and play. Maybe there's been a bit where the crowd and the players haven't been quite matched up, partly because of the hard times they've had. I think now there's beginning to get better with the players and supporters again, the crowd tonight was magnificent - it really was."
On Illiman Ndiaye's injury and Abdoulaye Doucoure's sending off: "They're going to check, they think he's maybe opened his knee but we're not sure yet. That was the feeling at the time that he's done something to his knee.
"I'm disappointed with Doucoure because we need him, we're down on numbers at the moment. We've got to be careful what we do. Discipline is never easy sometimes in these games and tonight was a game where quite easily sending offs happen."