Hengsbach obtained his doctorate in theology in 1944 from the University of Münich, with a dissertation entitled Das Wesen der Verkündigung - Eine homiletische Untersuchung auf paulinischer Grundlag.

He received his episcopal consecration on the following 29 September from Archbishop Lorenz Jäger, with Bishops Wilhelm Weskamm and Friedrich Rintelen serving as co-consecrators, in Paderborn Cathedral.

Hengsbach also served as Grand Prior of the Equestrian Order of the Holy Sepulchre of Jerusalem, and was made President of the German Episcopal Commission for Universal Church Affairs in 1976.

Pope John Paul II created him Cardinal Priest of Nostra Signora di Guadalupe a Monte Mario in the consistory of 28 June 1988.

Hengsbach lost the right to participate in any future papal conclave upon reaching the age of eighty on 10 September 1990 and, after a period of thirty-three years, resigned as Bishop of Essen on 21 February 1991.