According to this:
In the 1990s, the Japanese began incorporating computers into the animation process. Some works such as Ghost in the Shell and Princess Mononoke mixed cel animation with computer-generated images. Towards the late 1990s, companies had began shifting towards drawing cels digitally instead of with paint. Fuji Films to boldly announce the halt of cel production for the animation industry prompting a mass scramble to import foreign cels and transfer more of the production line to digital.
Princess Mononoke was released in 1997, though animation production began in 1995. Ghost in the Shell was released in 1995. Based on that, digital production as a part of the general production began in or before 1995.
According to this, which may or may not be correct (I don't know how reliable the source is), Tobira o Akete was an early anime short to feature digital production, as was Bit Cupid (an anime series rather than a short), both of which were from 1995.
Production I.G., known for their world-famous Ghost in the Shell film and other prestige features (Patlabor 2, Jin Roh) produced the first two digital anime series that most anime fans can recognize by name: Love Hina (with Xebec) and FLCL (with Gainax)
Final Fantasy: The Spirits Within, released in 2001, was the first photorealistic computer animated feature film. Blood: The Last Vampire, which was from 2000, was fully digital, and, according to this, the first fully digital feature.
The first fully digital anime series was Bit the Cupid, which was created in 1995 by Satelight Inc. A description is on this page, but it is in Japanese. From what I can tell from the translation, Bit the Cupid was the first continuous CG animation in the world. It was made to look like it was modeled in 3D. Also, after being colored, the contour lines were taken off.
All in all, it's kind of difficult to determine exactly when things began, but from what I can find, digital production began in 1995 and the first fully digital feature was released 2000.