United have already lost four league games this season and are currently ninth in the table.
Moyes said: “We have got a long way to go. We will need to play some great football and win an awful lot of games in the run-in if we going to be in there and around it at the end of the season.”
When asked if they can win the title, he replied: “Yeah, I’ve got nothing to suggest anything else.”
Moyes, who never won at Old Trafford during his 11-year reign at Everton, saw Roberto Martinez’s side claim all three points when left-back Bryan Oviedo scored an 85th minute decider in a 1-0 victory.
Moyes admitted it was really painful to lose to them, saying: “I know they are a very good team, Everton, with very good players so you would imagine it is a sore one when it is your old team. That is the way it is.”
Moyes was treated mercilessly by the travelling Everton fans, who chorused “David Moyes is full of s***” and “you’re getting sacked in the morning”.
However, he refused to make criticisms concerning them, adding: “It is not important. Everton supporters supported their team well tonight. They had to come here and make sure they got right behind their team and they did that.”
Wayne Rooney collected his fifth yellow card of the domestic season and will be suspended for Saturday’s home game against Newcastle.
His usual strike partner Robin van Persie was missing for a fourth successive match and, when asked if the Dutchman will return against Alan Pardew’s side, Moyes said: “Don’t know. We have been trying to get him back for a few weeks but he is not ready yet. It is nothing desperately serious, just a groin injury.”