A slow start to the 2013/14 campaign led many to write off the reigning champions, with a summer of change taking longer than anticipated to settle down.
Moyes has acknowledged that it was never going to be easy for him to pick up the baton put down by the legendary Sir Alex Ferguson and run with it, with time required to put his own stamp on things.
He has started to do just that and feels United are well placed to push on into the New Year, with the fierce competition which exists in the fight for the top flight crown leaving the Red Devils just eight points off the pace at the halfway mark.
Moyes said ahead of a meeting with Tottenham Hotspur on Wednesday: "We're definitely in it.
"But we've got a long way to catch up. We're not breathing down the necks of the teams above us at the moment. We've got to try to get a wee bit closer because, once we get in that position, people will know exactly what Manchester United are capable of.
"Right from the start I said it was going to be an unpredictable season, with so many changes at the top clubs. Probably the stability at Arsenal has given them that continuity that has allowed them to start the season well.
"But I've not changed my opinion - I still think it looks like it's an interesting season."
Six straight victories in all competitions have helped to keep United competing on multiple fronts, with Moyes seeing his squad respond positively to the questions asked of them in the early stages of the campaign.
The Scot added: "I think they've needed a challenge. The challenge is now, 'Well, we're not in the position we should be, let's get back to where we want to be'.
"I do think they've got the bit between their teeth at the moment and I want that to continue week-in, week-out.
"I don't think anyone's jumping up and down because we've won six games in a row. It's what's expected here and not all of them have been in the Premier League.
"We've still got a bit to do. Generally, I want us to play better. I want to make the team perform more consistently. That's where I have to get to."