He made a professional debut with the senior team on 11 September 2014 in an away game against Metalurgi Rustavi as a second-half substitute.
[7] In 2015, Zivzivadze moved to the top-tier club Samtredia, where he displayed his goalscoring abilities and played a crucial role in winning their first ever champion's title.
[8] Budu signed with Danish Superliga club Esbjerg fB in January 2017, but did not play much until he was loaned out to Dinamo Tbilisi for the 2018 season.
[12] Zivzivadze was named by Erovnuli Liga as the best player of part 2 of the four-phase championship after netting eight goals in nine games.
[16] During his second season at Karlsruher, Zivzivadze netted twelve goals in 30 league matches and became the team's second topscorer after Igor Matanović.
The player improved his own record in 2024–25 with the same number of goals scored in 17 games, which made him the league topscorer at half season.
[20] On 21 March 2024, Zivzivadze bagged a vital brace in a 2–0 win over Luxemburg in UEFA Euro 2024 qualifying play-offs,[21] which eventually helped the team to reach European Championship for the first time in their history.