Witnesses said the fire appeared to have started by an exploded projector bulb in the basement of the building, built between 1897 and 1909 in the Art Nouveau style and designed by Charles Rennie Mackintosh. The building is considered his masterpiece.
It is one of Britain's best-known buildings, and regarded as a national treasure.
Although the building was occupied by students preparing for final exams and an art exhibition, there were no injuries, Fire Officer Alasdair Hay said, adding the main fire was extinguished by late Friday but pockets of flames remained.
The Scottish Fire and Rescue Service said some of the objects within the building, including art works and Mackintosh-designed furnishings, were salvaged.
Witnesses reported seeing flames rise through the building and appear on the roof of the five-story building.
Austin Yuill, who is employed at the building, noted, "Quite apart from being voted the best building (in Scotland) of the last 175 years, it is a major tourist draw and has an incredible reputation as an art school. This is really terrible."