With both Manchester City and Chelsea dropping points over the weekend, Moyes believes that it will be a tight run to the finish line.
"I don't think anyone's going to run away with it," ESPN quotes Moyes as saying. "I think there are a lot of ups and downs, it's a close-run league this year.
"I think there will be a few shocks, and we've think we've had one or two at the early part of the season, but I'm pleased we've been able to show a little bit more consistency and a bit more momentum. Let's hope we continue that.
"Manchester United were champions last year and there's a lot of people written them off very quickly, which is fine, but hopefully, come the end of the season, we will not be too far away from it."
As a result of yesterday's win, United have moved up to fifth in the table.