I know that original animes weren't based on light novels or mangas (see: this). However, I think there are some early shows that are based on mangas (Astro Boy for example, I think). What was the first anime based on a manga? Also, what was the first anime based on a light novel?
While not the most scientific method, one could generate a list of anime entries over at Anime News Network, sort it by order and see what is the earliest entry in their database that fits the criterion we're looking for. Here would be an example of such a list.
Obviously the service's database is most likely incomplete, but regardless, it is still an authoritative source for anime information, and given the size, and time range of the database, it can be trusted as a good estimation.
Technically, Tetsuwan Atom (1963) is the first anime to exist and was also based on a Manga, but if you do not get too technical on what anime means and just refer to animation from Japan you get Shou-chan no Bouken (1924) as the first such "anime" that was based on a "manga".
The first anime based on a Light Novel is Kikansha Yaemon: D51 no Daibouken (1974) and the first "anime" based on a normal novel is Saiyuuki: Son Gokuu Monogatari (1926).
Though, what we commonly refer to as anime nowadays - the whole subculture stuff - only originated around 1978-1982, so none of the titles above fit if you go by that understanding of anime.