Police in Bavaria previously said they suspected the fire was arson.
The alarm was raised at 9 p.m. on Friday, but because of the steep path up the mountain, the local rescue service did not reach the top of the alp above Berchtesgaden until 9:52 p.m.
Police have now arrested a man suspected of starting the fire.
The building, called the Kehlsteinhaus, stands at an altitude of 1,834 meters in Obersalzberg, near the Austrian border, and is a popular tourist destination.
The house was built for Adolf Hitler in 1938 on the orders of Nazi leader Martin Bormann, but Hitler rarely visited the Kehlsteinhaus. He preferred to stay farther down the mountain, at the sprawling Berghof complex, which served as the Nazis’ second government headquarters.