In 1950, Norakidze signed for Dinamo Sokhumi, moving to Spartaki Tbilisi the following year, scoring eight goals in 25 appearances in the Soviet Top League.
In 1958, Norakidze moved back up to the Soviet Top League, signing for Dinamo Tbilisi, making five appearances in the competition.
In 1969, Norakidze had a short stint as manager of Meshakhte Tkibuli, before moving to Dila Gori the same year.
After departing Torpedo Kutaisi, Norakidze was appointed manager of former club Dinamo Sokhumi.
In 1983, Norakidze joined Torpedo Kutaisi for a fourth time as manager, departing later that year to rejoin Lokomotiv Samtredia.