Born in Wiesbaden, Rhenish Hesse, Unverzagt studied classical philology, archaeology, and geography at the universities of Bonn, Munich, and Berlin between 1911 and 1914.
He then worked for a short time as a research assistant at the Museum of Nassau Antiquities in Wiesbaden and from December 1916 to Summer 1917 in the Römisch-Germanische Kommission of the German Archaeological Institute in Frankfurt am Main.
He was then employed by the military in Brussels as an assistant consultant on the staff of the Flemish Occupation Administration, where he recorded Roman and late antique monuments in Belgium and northern France.
Due to his efforts to rescue Belgian and northern French artworks, he was first appointed to the German Armistice Commission in Spa, Belgium, became a consultant from January 1920, and then worked in the Reich Commissariat for Reparations in Berlin.
Following these years of wartime interruptions Unverzagt resumed his studies and received his doctorate on 3 March 1925 from the University of Tübingen with the classical archaeologist Carl Watzinger.
According to the Akinsha and Kozlov, Unverzagt "knew it was better to give the art treasures to the enemy, who would probably return them eventually, than to let thieves and pillagers take them."
Already in 1927 he had been instrumental in founding the "Working Group for the Study of Pre- and Early Historic Hill and Defensive Fortresses in Central and Eastern Germany".
In 2004, the Museum also acquired the scientific documents and materials that were found in Unverzagt's apartment in Berlin-Charlottenburg after his death, and assembled them for a partial retrospective.