Kagawa is set to complete his move from Borussia Dortmund in the next few weeks, once he passes a medical and obtains a work permit.
He will become United’s first-ever Japanese player and Zaccheroni believes the midfielder can handle the responsibility.
The Italian said: "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.
"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 himself is yet to comment on the proposed transfer, but Zaccheroni says it augurs well 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 and now Kagawa to United. It's proof the Japanese are developing players the right way."