The Toffees chief watched his side demolish Wigan 4-0 at Goodison Park on Sunday thanks to a brace from on-loan striker Jo and one each from Marouane Fellaini and Leon Osman.
And, while a place in next season's new Uefa Europa League is virtually in the bag, Moyes is still looking at Champions League qualification.
Everton are a point behind fifth-place Aston Villa and seven points behind Arsenal in fourth spot. However, with seven games still to play Moyes does not believe that gap is insurmountable for his side.
He said: "The Champions League is still there for us, we are not going to give up.
"Arsenal have got the points and are coming into form with players back from injury.
"We are not going to give up, but realistically it will be a tough call to catch them now.
"We have got the defeat at Portsmouth out of our system. I was worried about that, but we were back to our best.
"It was a good performance, a good result and another clean sheet. I was pleased with much of what I saw."
Everton, who are also still in the hunt for FA Cup glory as they face Manchester United in the semi-finals later this month, have got a massive away game at fifth-placed Aston Villa next Sunday. And punters can use a £10 free bet if they feel the Toffees can win that game and go on to pinch a top-four finish from under the noses of Arsenal.