Zhdanov was born in the village of Shtepino (in present-day Donetsk Oblast, Ukraine). After Zhdanov graduated from Kharkiv Medical Institute in 1936, he spent the next decade working as an army doctor, where he became interested in epidemiology; this work would directly lead to his doctoral thesis on Hepatitis A. In 1946, Zhdanov was invited to become Chief of the Epidemiology Department of the I. I. Mechnikoff Institute of Epidemiology and Microbiology in Kharkiv, becoming its director two years later. His work in virus classification saw him admitted to the International Committee on Taxonomy of Viruses as a life member.
In 1958, Zhdanov, as Deputy Minister of Health for the Soviet Union, called on the World Health Assembly to undertake a global initiative to eradicate smallpox. The proposal (Resolution WHA11.54) was accepted in 1959. Zhdanov left the Ministry of Health in 1961, and focused on scientific research for the rest of his career. This work included studying influenza, hepatitis, and in the 1980s, HIV.
In addition to his accomplishments in the field of public health, Zhdanov chaired the Soviet Union's Interagency Science and Technology Council on Molecular Biology and Genetics, which among its many functions directed the Soviet biological weapons program.