Bronsart was born in Danzig (present-day Gdańsk, Poland), then administrative capital of West Prussia, the son of later General Lieutenant Heinrich Bronsart von Schellendorff (1803–1874) and his wife Antoinette Martha Elisabeth, née Drège (1810–1873). His younger brother Walther Bronsart von Schellendorff (1833–1914) also served as Prussian Minister of War from 1893 to 1896.
On 30 May 1853, Paul Bronsart married Rosalie Klara Marie, née Schmidt (1833–1913). The couple had four children, among them Fritz Bronsart von Schellendorf (1864–1950), General lieutenant and chief of the Ottoman Army general staff in World War I.
Having attended the Gymnasium in Danzig, Bronsart entered the Prussian Cadet Corps (Kadettenanstalt) in Kulm and Berlin. On 28 April 1849 he joined the Berlin Kaiser Franz Garde-Grenadier Regiment No.2 in the rank of lieutenant. From 1 June 1852 he served as adjutant in the 4th Guards Landwehr Regiment and attended the Prussian Military Academy from 1855 to 1858. On 1 May 1859 he was appointed to the German General Staff and achieved the rank of Oberleutnant. Elevated to Hauptmann (Captain) on 23 February 1861; after three years of staff service he returned to regimental duty as a company commander in the II Army Corps, but was soon reappointed to the staff, and lectured at the Military Academy, becoming major in 1865 and lieutenant colonel in 1869.
Bronsart participated in the Austro-Prussian War of 1866 as a member of the II Corps general staff, and fought in the Battle of Gitschin and Königgrätz. During the Franco-Prussian War of 1870/71 he was chief of a section on the Great General Staff headquarters, and conducted the preliminary negotiations for the surrender of the French at the Sedan fortress. After the war Bronsart was made Oberst (colonel) and Chief of Staff of the Guard Corps, becoming major general in 1876, commander of the 1st Guards Infantry Brigade in 1878, and lieutenant general (with a 2nd Guards Infantry Division command) in 1881.
In March 1883 he was appointed Prussian War Minister, succeeding Georg von Kameke. During his tenure of the post, until 1889, the peacetime establishment of the German Army was significantly increased and many important reforms were carried out in the Prussian Army, in particular the arming of the infantry with repeating rifles as well as new regulations of pension funds and compulsory military service. On 23 April 1888 he was promoted to General of the Infantry.
Minister Bronsart resigned on 8 April 1889 and was succeeded by Julius von Verdy du Vernois. At his request, he was appointed commanding general of the I Army Corps at Königsberg. Two years later he died from pneumonia at his estate of Schettnienen near Heiligenbeil (present-day Mamonovo, Kaliningrad Oblast) at the age of 59.