Born in
Gudauta, Tsveiba began playing youth football with local side M. Chachba. He played as a defender for
Dinamo Sukhumi,
SKA-Khabarovsk,
Dinamo Tbilisi and
Dynamo Kyiv, appearing in 123 Soviet league matches. Tsveiba won the
1990 Soviet Top League with Dynamo and was awarded Master of Sports of the USSR. Tsveiba started his career at Dinamo Sukhumi before playing regularly for Dinamo Tbilisi. In 1990, he left for Dynamo Kyiv, where he was nominated by the
Ukraine national team. He then played briefly in Russia before signing with
Gamba Osaka. He returned to Russia for
Alania Vladikavkaz, where he was nominated by the
Russia national team. Tsveiba left for the Chinese team
Shanghai Pudong, before spending his late career with the Russian teams
Uralan Elista and
Dynamo Moscow. In 2009, he was part of the Russia national team squad that won the
2009 Legends Cup. ==International career==