The Toffees are virtually guaranteed to be playing European football next season, with a 3-1 win over Stoke on Saturday strengthening their grip on sixth spot in the table.
However, Moyes will not allow his players to slip into a comfort zone with nine games remaining.
With only four points separating the club from a UEFA Champions League place, the ambitious Scot has not given up hope of gate-crashing the top four.
"We are still going to chase the teams above us; we are going to keep those below away from us and we are going to try to reach a cup final," Moyes vowed.
Effort"This is the best squad I have had at Everton, and it is being underlined by the effort the players are putting in.
"The quality we have is excellent - if not the quantity of players in the squad. Nevertheless, we are still getting good results."
Everton centre-half Joleon Lescott is also keen to avoid any unnecessary slip-ups between now and the end of the season.
"Every game is vital now; every team is fighting for something, so you must go into every game showing your opponents respect regardless of where they are in the table," said the England international.
"But there is a great momentum around our club at the moment - we have to keep going to achieve our aims of qualifying for Europe and getting to the (FA) cup final."