After finishing secondary school in 1925 in Passau, Peschl started studying mathematics, physics, and astronomy in Munich.

He received his doctorate in 1931 from the University of Munich under the supervision of Constantin Carathéodory with a dissertation titled Über die Krümmung von Niveaukurven bei der konformen Abbildung einfachzusammenhängender Gebiete auf das Innere eines Kreises; eine Verallgemeinerung eines Satzes von E. Study ("On the curvature of level curves of the conformal mapping of simply connected domains to the interior of a circle: A generalization of a theorem of Eduard Study").

Under pressure Peschl became a member of the Nazi Party and the paramilitary Sturmabteilung, but he avoided any activity within either organization and ended SA service after a year.

From 1943 to 1945, he worked at the German Aviation Research Institute in Brunswick, which exempted him from further military service during World War II.

Peschl was the doctoral advisor of Claus Müller, Friedhelm Erwe, Karl Wilhelm Bauer, Bernhard Korte, Stephan Ruscheweyh, and Karl-Joachim Wirths, among others.