Eliava was born into an impoverished noble family from western Georgia. He studied law at the Petersburg University in 1903, but was excluded for his involvement in students’ protests. In 1904, he joined the Bolshevik wing of the Russian Social Democratic Labour Party and took part in the 1905 uprisings in Tiflis and Kutaisi. In 1911, he was sent into exile to the Olonets Governorate whence he moved to Petersburg and worked for the Pravda newspaper until being deported to Vologda in 1915. After the outbreak of the Russian Revolution of 1917, he chaired the Vologda Ispolkom ("executive committee") and moved to Moscow in December 1918. In 1919, during the Russian Civil War, he was sent to the Turkestan front and, as a member of the Revkom ("revolutionary committee"), helped establish Soviet power in the region. In 1920, he served as Soviet Russia’s plenipotentiary in Turkey and Iran. During the war with the Democratic Republic of Georgia (DRG) early in 1921, he was a member of the Caucasus bureau of the Russian Communist Party, and a member of the Georgian Revkom. After the overthrow of the DRG, Eliava joined the government of the Georgian SSR as a People's Commissar for Navy and Military. From January 1923 to June 1927, he was a head of the Soviet Georgian government as a Chairman of the Council of People’s Commissars of the Georgian SSR. Simultaneously, from 1927 to 1930, he served as the Chairman of the Council of People’s Commissars of the Transcaucasian Socialist Federative Soviet Republic. In the 1930s, Eliava was a deputy People’s Commissar of Foreign Trade and a deputy People’s Commissar of Light Industry of the Soviet Union. He was arrested during Stalin's crackdown on Old Bolsheviks and executed in December 1937.