Leselidze was born on October 2 or 15, 1903 in Ozurgeti, Georgian (then part of the Russian Empire) to an ordinary employee family. He was the elder brother of another decorated warhero, colonel Victor Leselidze who also died in the same year of the war. In May 1921 after graduating from high school in Tbilisi Leselidze joined the Red Army and was deployed primarily in Georgia. He participated in the suppression of local anti communism movements and uprisings before graduating from the Georgian Joint Military School in 1925 and in 1929 from the Tbilisi Artillery School for advanced officer courses. Leselidze was member of the All-Union Communist Party Bolsheviks already since 1925. During the period of 1922 to 1938 he commanded artillery units from batteries to regiments and the Georgian Joint Military School. In June 1938 he got appointed to chief of artillery, infantry division and was active during the Soviet invasion of Poland in 1939. From February 1941 to the outbreak of hostilities, colonel Leselidze was chief of artillery, rifle corps in the Belorussian Special Military District.