District chief Somchai Traithipchartsakul said that 93 shelters in zone C had been razed and 595 Karen refugees living there had been affected.
Mr Somchai said camp and district officials worked quickly to contain the fire and keep it from spreading to other zones.
Only one person suffered burns but was not seriously injured. The man was sent to a nearby hospital for medical treatment.
Mr Somchai said soldiers from the Third Region Army, Tha Song Yang district officials, the Tak Red Cross and NGO officials had distributed necessities including food, water, clothes, blankets and medicines to the affected refugees.
Construction of new shelters would begin within the next two days, he added.
The affected refugees are now staying with their relatives inside the same camp.
Mae La is the largest of the nine refugee camps along the Thai-Myanmar border, which altogether house around 150,000 refugees. It is home to 53,000 people, 90% of them Karen, some of whom have lived at the camp for 30 years.