Following their 0-0 draw at Arsenal on Tuesday, the Toffees are now two points behind Chelsea and Tottenham Hotspur having played two games more than the Blues and one more than the North Londoners.
The Scot feels that only maximum points from their final five fixtures of the Premier League season will give them a chance of finishing in the top four.
"We probably need to win all our games to have any chance but the players are playing well, we've had a couple of really hard away games and we keep picking up points," Moyes told Sky Sports News. "I thought it was a really good game, it was end to end, we played some good stuff and we had to defend well.
"It was never going to be easy for us but I'm not going to be too despondent with a point even though I wanted to win it. We got a little bit fortunate and there were a couple of bits of fantastic defending to make sure we didn't concede a goal.
"We didn't have as many chances as we would have liked in the second half, but any manager who comes here knows that you're not going to get loads of opportunities."
Everton finish the campaign with games against Sunderland, Fulham, Liverpool, West Ham United and Chelsea.