The Toffees claimed a point from a battling 0-0 draw with fellow top-four hopefuls Arsenal at Emirates Stadium on Tuesday night to remain sixth in the table.
Everton are now two points adrift of Tottenham in fifth and Chelsea in fourth, but have played a game more than Spurs and two more than the Stamford Bridge side.
Moyes' men finish the campaign with games against Sunderland, Fulham, Liverpool, West Ham and Chelsea, with the Scot feeling only maximum points will see them secure a place among Europe's elite next term.
He told Sky Sports: "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.
"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."