The 18-year-old midfielder joined Arsenal in January and was loaned to Dutch side Feyenoord as he was unable to obtain the necessary paperwork having not represented Japan's senior side.
An FA panel on Tuesday agreed to grant the documentation after hearing from Arsenal manager Arsene Wenger and reading a letter of recommendation from the Japan Football Association.
"We are absolutely delighted... this is almost like a new signing for us," Wenger told Arsenal's official website.
"Ryo has worked extremely hard in pre-season and performed very well during the matches in Asia, Germany and Portugal.
"Ryo has natural technical ability. He has good balance and phenomenal pace, and his passing and crossing is consistently of a high quality."
The Frenchman added: "He is a very exciting player. We're looking forward to his contribution this coming season with Arsenal."
Miyaichi follows countryman Junichi Inamoto, who had a spell at Arsenal from 2001-02 but saw little first-team action and was affectionately dubbed "T-shirt" by the home fans, in reference to his commercial value to the club in Japan.
Miyaichi, who helped Feyenoord avoid relegation from the Dutch first division last season with three goals in 12 matches, will be eligible to play for Arsenal once the formalities of his paperwork are completed.
Arsenal play Newcastle United away on Saturday in their opening match of the 2011/12 Premier League season.
