The Dutchman, who currently leads the Premier League scoring charts with 17 goals, sealed a 30.5 million euro move to Old Trafford over the summer after eight years in north London.
The 29-year-old has now stated that he felt he had reached a point where he required a "new challenge" away from the Gunners.
"Arsenal are a great club and I always had a good feeling but after eight years, sometimes you feel that you need a new challenge," Van Persie told Sky Sports. "I found that challenge at Manchester United. I do feel that I am surrounded by champions. I do feel that this is exactly the right time to be here. I don't think it would have been better to come here earlier. Everything has gone this way for a reason.
"When I extended my deal a couple of years ago at Arsenal, I felt that was the right move for me - to stay and play because I wasn't finished there. Now, since last summer, I wanted to go for a big challenge, and this was the biggest challenge for me.
"I think every day tells me that I made the right move. It's nothing bad about Arsenal. No player is bigger than the club, Arsenal will always challenge for trophies and be up there. It was just that after eight years I was ready to move on and try something else."
The Rotterdam-born forward also spoke of the ease with which he has settled in at United and revealed the exciting nature of training with his new teammates.
"On and off the pitch, everyone has been helpful. I have settled in well," he continued. "The main thing is that from day one everyone gave me the feeling that I was welcome and that they were happy for me to be here. I was happy to be here, it was a new challenge for me and I was ready for that challenge. I am having fun.
"I am seeing things again which I haven't seen for a long time. When I first went to Arsenal years ago I saw things from [Dennis] Bergkamp and [Thierry] Henry that I had never seen in my life before. Now I am seeing those things again and that makes me happy. It's like being a kid in a sweet shop. This is having fun, enjoying myself. That's what I am doing, I'm just having fun."
Van Persie went on to insist that his previous struggles with injury were down to bad luck rather than any long-term physical problems.
"If you go back to my injuries, people just look at stats and for a couple of years those stats weren't great. But most of those injuries were impact injuries, they can happen. It was mostly unlucky," he added. "I don't think it was me to blame, or the club medical staff to blame. I was just unlucky."