Asai was born to an ex-samurai class household in Sakura, in the Kantō region of Japan, where his father had been a retainer of the Sakura Domain. He attended the domain school, where his father was principal, and left home in 1873 to pursue English language studies in Tokyo. However, he became interested in the arts, and enrolled as a pupil of Kunisawa Shinkuro in western oil painting classes. In 1876, he enrolled as one of the first students in the Kobubijutsu Gakkō (the Technical Fine Arts School), where he was able to study under the Italian foreign advisor Antonio Fontanesi, who had been hired by the Meiji government in the late 1870s to introduce western oil painting to Japan.

In 1889, he established the Meiji Bijutsukai (Meiji Art Society), the first group of Western-style painters in Japan, and in 1898, he became a professor of the Tokyo School of Fine Arts (present day Tokyo University of the Arts. However, in 1902 he resigned his post and travelled to France, where he spent the next two years refining his techniques in the impressionist school.