United were crowned Premier League champions for a 13th time in 2012/13, with the Red Devils dominating the English top flight.
Van Persie led the title charge in his first season at Old Trafford and landed the golden boot as he found the target on 26 occasions.
A summer of change at United has seen Sir Alex Ferguson slip into retirement and David Moyes succeed his fellow Scot, but there is a feeling within the camp that a seamless transition will be made in the pursuit of further top honours.
Van Persie said in the Daily Mirror: "I have a feeling that we can do much more and that we can win many more trophies.
"I've had a year to work with everyone and now the whole coaching staff is changing, which makes it a new challenge.
"I've got to know the players through and through and I've a feeling they have a lot more in them. A whole lot more.
"The season began as a nice challenge but you never know how these things will turn out."
On Ferguson's decision to call time on a 27-year stint in the Old Trafford hot-seat, Van Persie added: "I'd have preferred it if he'd stayed on a few more years. But I understand his decision.
"I'm even prouder that I could work under him for a year. I could see he was a legendary man."