United, down to ninth, are now 12 points behind table-toppers Arsenal with both teams having played 14 league fixtures.
When asked about that gap, 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 come the end of the season."
Then asked if he thought his team was capable of doing that, he said: "Yeah - I've got nothing to suggest anything else."
In his 11 years as Everton manager prior to his defection to United over the summer, Moyes had never overseen a win at Old Trafford.
And after witnessing his successor Roberto Martinez pull off the feat, with the Toffees sealing their first away victory over the Red Devils since 1992 courtesy of full-back Bryan Oviedo's 85th-minute strike, Moyes said: "I'm disappointed to lose but that is the way the game goes sometimes.
"The game was tight. We missed some opportunities to score and Everton took it when they got their opportunity."
Asked if the loss hurt him more than usual as it was to Everton, he added: "It doesn't hurt any more.
"I know they are a very good team, Everton, with very good players.
"You would imagine it is a sore one when it is your old team. That is the way it is."
Despite his years of service to Everton, the club's fans gave Moyes - who made a joint bid that was deemed derisory by the Toffees for Marouane Fellaini and Leighton Baines over the summer, and eventually secured the former for United - a hostile reception during the match.
And asked about that, the Scot said: "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."
In contrast to the testing time Moyes - whose team have now suffered four top-flight defeats in 2013-14 - is enduring, it seems things could hardly be going better for Martinez in his new job.
Everton have been beaten only once in the league this season and currently lie fifth in the table, three points off second place.
And on the Toffees ending their Old Trafford hoodoo, Martinez said: "Today was more than a football game for us - it was about trying to get over the mental block that we have had for many, many years.
"Someone told me was 20-odd years since we won at Old Trafford.
"It was about being ourselves throughout the game. I couldn't be prouder.
"The performance is not by any means our best of the season, but we had a real mentality in the first half when we had to dig in, when Manchester United created a few chances as you would expect.
"We never felt inferior.
"I have always felt this squad is capable of going anywhere in this league and playing teams eye-to-eye.
"In the first half we did certain good things. The second-half performance was a lot better and when we had to defend, we did, when we had to keep the ball, we did, and when we went into the final third we had real composure and looked a real threat.
"I told the boys afterwards that this gives a real indication of what can be achieved.
"But we have done nothing - we need to carry on improving, carry on getting better."
The Spaniard was reluctant to speculate about where his team might finish this term, but was keen to praise Oviedo, who now has two goals from the two starts he has made since coming into the team for the injured Baines.
Martinez said: "I think Bryan has got the hardest job in football - to be fighting against one of the best left-backs in world football, who is in one of the best moments of his career, is quite difficult.
"It would be easy to just drop your standards and be a little bit of a nuisance.
"But Bryan has been working hard, waiting for his chance and what he has done in the last two games is a real example of how a footballer should behave."
Meanwhile, Moyes said he did not know when injured United striker Robin van Persie, missing for the last four games, would be back in action.
United will definitely be without Wayne Rooney for Saturday's home league clash with Newcastle, with the frontman's booking in the Everton match being his fifth for the Red Devils of the season and earning him a suspension.