Moyes told evertonfc.com: "There will always be eyebrows raised (at team selection) and you are going to have to play well to be in the team. That goes for everybody.
"I thought Leighton played well and in a lot of ways I didn't think we backed him up in his performance."
He added: "Leighton will be pleased to get that one under his belt and hopefully next time he comes and plays for us we will play much better.
"Phil (Jagielka) did make a difference (when he came on), that is all credit to him.
"We have tried to break him in carefully, like we did with Joleon (Lescott) when he was coming off the bench in the first few games last season."