The Old Trafford side suffered a fourth defeat of the season when they were stunned 1-0 at home by his former club Everton on Wednesday.
The champions are now ninth in the table and are as close to the relegation zone as they are to the summit.
United only lost five league games in the whole of last season, but even though Moyes accepts it will be a struggle, he insists his team should not be ruled out.
“We have got a long way to go,” he said. “We will need to play some great football and win an awful lot of games in the run-in if we are going to be in there and around it at the end of the season.”
Asked if he believed United were capable of that, he replied: “Yes. I’ve got nothing to suggest anything else.”
The United manager admits he does not know when last season’s top scorer Robin van Persie will return from a groin problem.
Striker Wayne Rooney, meanwhile, will be suspended for Saturday’s home match against Newcastle United after picking up five yellow cards.
Netherlands international Van Persie has not played since the 1-0 win over Arsenal on November 10.
“It is a sort of a groin injury. Nothing desperately serious, just a groin injury,” Moyes said.
“We have been trying to get him back for a few weeks, but he is not ready yet.”
Moyes was the target for chants from Everton fans angry at the manner of his departure during the close season and his subsequent attempts to sign Marouane Fellaini, who joined in a £27.5 million deal ($45 million, 33.2 million euros), and Leighton Baines.
But the Scot, who spent 11 years at Goodison Park, said that he had no issues with the supporters of his former club.
“It is not important. Everton supporters supported their team well,” he said.
“It (defeat) doesn’t hurt more being Everton. I know they are a very good team with very good players. You would imagine it is a sore one when it is your old team, so that is the way it is.”
After Rooney and Danny Welbeck hit the woodwork either side of half-time, Everton’s Costa Rican left-back Bryan Oviedo turned in the game’s only goal from Romelu Lukaku’s scuffed shot with four minutes remaining.
Everton are fifth in the table and five points above United after their first victory at Old Trafford since 1992.
Moyes’s successor, Roberto Martinez, believes that the victory at United will help his team change their mindset against the Premier League’s biggest sides.
“It was more than a football game; for us it was trying to get over the mental block we had for many years,” said Martinez.
“Someone told me it’s 20-odd years since we won at Old Trafford, so it wasn’t about three points. It was about being ourselves throughout the game and pushing each other.
“The performance isn’t the best this season — we’ve played a lot better — but we had a real mentality when we had to dig in and they created a few chances.
“But we never felt inferior and kept doing what we’re good at, created many open chances, which is always hard to do, and overall I couldn’t be prouder of the mentality of the players.”
On Oviedo, who has been used as back-up to Baines since joining Everton in 2012, Martinez added: “Bryan has got the hardest job in football to be fighting against one of the best left-backs in world football at the peak of his career.
“It’s reward for his hard work and he deserves it.”