In 17 years he started playing for the "Dynamo" (Sukhum). At the gathering at the Olympic base in Esherf he was noticed by coaches of SKA (Khabarovsk), for which he played for 3 months.

He made his debut in Tbilisi in 1985 in a match against "Kairat". The first success as a player began to seek in the "Dynamo" (Tbilisi). In late 1989, on the advice of Lobanovsky, he moved to the "Dynamo" (Kiev).

Akhrik was invited in late 1991 on "Tottenham Hotspur", and representatives of the English club has traveled to the airport to meet Tsveiba, but he refused.

Tsveiba expected to leave after Euro 92, but standing offers was not, and he continued to play for the "Dinamo". In the winter season 1992/93, he traveled together with Oleg Salenko on view in "Karlsruhe".

In the middle of 1993 he was in Russia, where he performed for Kamaz. In 1994 he went to Japan, where picked up a club personal agent. He played for the "Gamba Osaka", where his partners were Sergey Aleynikov and Oleg Protasov. In summer 1996 he played in the traditional match between Legionnaires J-League qualifier against Japan.

In January 1997, called for the Russian national team at the Commonwealth Club Cup 1997. A few weeks later he signed a contract with "Alaniya", in which he played until the end of the year.

Akhrik Tsveiba has finished his playing career in 2001 in the Moscow "Dynamo", this team became the fourth Dynamo in Tsveiba`s career. In 2003 he was appointed to the post of coach-breeder club "Torpedo-Metallurg" in Moscow. In recent years, in favor of the team of veterans of Russian football. Currently working as head of the breeding department of the Moscow "Dynamo".

Player for four national football team: the USSR, the CIS, Russian Federation and Ukraine (played one match in 1992).

Member of the European Championship in 1992.