Eberspächer was born on 18 November 1915 in Tübingen, at that time part of the Kingdom of Württemberg, a federated state of the German Empire. In 1934 he volunteered for military service in the Reichswehr, and served in an armoured reconnaissance battalion. In parallel to his military service he studied mechanical engineering at the University of Stuttgart. Following his graduation in 1939, he transferred to the Luftwaffe. From July 1940 to January 1943 he served with the 7.("Fernaufklärungs"-) "Staffel" of "Lehrgeschwader" 2 (7th Long Range Reconnaissance Squadron of the 2nd Demonstration Wing) on both Western and Eastern Fronts. He then converted to a ground attack role, and from March 1943 served with "Schnellkampfgeschwader" 10 (SKG 10—10th Fast Bomber Wing), flying fighter-bomber missions over southern England and anti-shipping operations in the English Channel. He was promoted to the rank of "Hauptmann" (captain) on 1 May 1944 and appointed as "Staffelkapitän" (leader of the squadron) in Kurt Dahlmann's I./SKG 10 (1st Group).