Kagawa is set to sign for Sir Alex Ferguson's side in the coming weeks after United agreed a deal for the Borussia Dortmund midfielder.
The proposed move would see him become the first Japanese player to don the famous red jersey.
Although some sections of United's fans had been hoping for a bigger-name signing, Zaccheroni is confident the 23-year-old will eventually change their minds and blossom into an integral player for the Red Devils.
"I haven't offered him my congratulations yet. I've only said to him 'good luck' because this is just the start for him. I know he'll need results to be deemed a success at Manchester United," the Italian told United's official website.
"But I fully expect to congratulate him three years from now because he clearly has the mental strength and the technical ability to make it there."
Kagawa was tight-lipped when asked to comment on the impending transfer three days ago, but Zaccheroni sees the move as a sign of change for Japanese football.
"I know he doesn't want to talk about it, but I also think it's an issue you can't ignore either," he said.
"I think his transfer is great not only for himself but for all football in this country.
"I've been in Japan for almost two years now and in that short period of time, two players have signed with the biggest clubs in the world - [Yuto] Nagatomo went to Inter (Milan) and now Kagawa to United. It's proof the Japanese are developing players the right way."