TOKYO—Japanese videogames are attracting a bigger U.S. fan base with growing appreciation for the distinctive look and storylines in the latest batch of games from the country’s developers.

Gree Inc., a Tokyo-based smartphone game company, said in April that its “DanMachi” role-playing game earned 40% more in the U.S. than in Japan—the first time that has happened for the 14-year-old company.

It...