The Toffees shared the spoils with Stoke in a 1-1 draw at the Britannia Stadium on Saturday, when a win would have seen them move level on points with third placed Chelsea. Tony Pulis' men are unbeaten at home in the Premier League this season and Moyes was full of praise for Stoke.
"We'll do everything we can to get into the top four," said Moyes.
"I don't know who's going to be in it. It wouldn't be a surprise if Stoke were pushing towards that top end.
"I think all teams will find it difficult at their place. They are very strong, very powerful and they are a good team.
"They need more credit. There was never only one way for football to be played. There are plenty of rules. You've got to be able to play against them."
However, Moyes did cast doubt over his own team's chances of qualifying for the Champions League, saying:
"I don't know, when it comes to it, if we will actually have enough this season. Realistically, will it happen? I don't know.
"But I want us to try and keep pushing and pushing and try and get a position in European football if we can next year."