Georgi Daneliya was born on August 25, 1930, in Tbilisi, Georgia. His family relocated to Moscow in 1939, where his father worked as the chief engineer for the Moscow Underground Metro System. Daneliya's father, a decorated World War II General, specialized in constructing underground bunkers for the Soviet Government. His mother, a skilled chess player, later worked as a second-unit director at Mosfilm. Daneliya's maternal aunt, Veriko Anjaparidze, was married to Mikheil Chiaureli, a personal friend of Joseph Stalin.
Daneliya initially earned his architecture degree from the Moscow Architecture Institute. He then studied at the Higher Director's Courses at Mosfilm under Mikhail Romm and graduated in 1959, becoming a film director at the Mosfilm Studios in Moscow.