In 2013, after having spent a single season with the reserve side of the Georgian club, Papunashvili was promoted to the first team and in November he made his debut in Erovnuli Liga against Tskhinvali.
[2] During the 2014–15 season, Papunashvili became the key figure at Dinamo, scoring 16 goals in all tournaments with the club, including his first career hat-trick against Tskhinvali in October 2014.
[6] The loan spell was unsuccessful, as Papunashvili missed 12 games due to injury and was only able to make 20 appearances for the club, scoring two goals.
[11] On 12 January 2020, Papunashvili joined fellow second division side Racing de Santander on loan until the end of the season.
[18] Papunashvili made his debut for the national team in a 1–0 friendly loss against the United Arab Emirates on 3 June 2014.