Kurashvili began aerobic gymnastics at the age of five.[2]
In December 2010, she became the youngest competitor of the Ukrainian Cup among cadets, finishing fourth.[6]
In July 2013, she won a Cup of Azov Sea, the all-Ukrainian aerobic gymnastics competition, held in Berdiansk.[7] In October, Kurashvili received a bronze medal in individual at the international aerobics competition, held in Plovdiv, Bulgaria among pupils.[8] In October 2016, Kurashvili won a gold medal at the Sumy Aerobic Cup in individual event.[9]
In 2018, Anastasiia competed at the World Junior Championships, held in Guimarães, Portugal, placing second in the individual with a score of 20.65.[10] In May, she competed at the World School Sport Games in Marrakech, winning a bronze medal in trio competition.[11]
The following year, at the European Junior Championships in Baku, Azerbaijan, Kurashvili won a bronze medal in the individual event.[12] In that year, she also won a bronze medal in the mixed pair at the 9th International Open Competition in Portugal.[13]
In May 2021, Kurashvili competed at the World Championships in Baku, finishing seventh in trio.[14] In September, she also took part at the European Championships in Pesaro, finishing 6th in individual and 7th in mixed pair and trio.[15]
During the Russian invasion of Ukraine in 2022, Anastasiia moved to Italy for further training and preparation for the international competitions.[16] In June, she won the gold medal in individual at the Italian Aerobic Championships, qualofying at the World Championships.[17] Later, Anastasiia finished first with a score of 20,950 at the World Championships in Guimarães.[18] Later, she competed at the 2022 World Games in Birmingham, where she finished 6th in trio competition with Stanislav Halaida and Anastasiia Lytvyn in qualification without reaching a final.[19]
In March 2023, at the FIG World Cup in Cantanhede Kurashvili won a gold medal in mixed pair and a silver one in individual.[20] In May, she was nominated for Female Gymnast of the Year 2022 according to European Gymnastics Federation.[21] In October, Kurashvili added to her personal award two silver medals in mixed pair and individual at the FIG World Cup in Bucharest, Romania.[22] Later, she was awarded the Best Acrobatic Gymnast of the Year according to FIG.[23] In November, Kurashvili competed at the 2023 Aerobic Gymnastics European Championships, where she won a silver medal in individual and a bronze one in team competition.[24][25]
In March 2024, at the FIG World Cup in Cantanhede Kurashvili won two gold medals in mixed pair and individual.[26] In May, she won two silver medals in the individual and mixed pairs events at the Suzuki Aerobic World Cup.[27] In September, she won a silver medal in individual event at the World Championships in Pesaro.[28]
In 2025, Kurashvili won gold and silver medals in the individual and mixed pairs (with Stanislav Halaida) respectively at the 10th Cantanhede FIG World Cup.[29] Later, she finished 5th in mixed pairs event at the FIG Suzuki World Cup.[30]
In August, she competed at the 2025 World Games in Chengdu with her partner Stanislav Halaida, winning a bronze medal in pairs competition, firstly in Ukrainian history.[31] In November, she won a historic gold for Ukraine in the individual event at the Aerobic Gymnastics European Championships in Ganja, Azerbaijan.[32] There she won a silver medal in mixed pairs competition with her partner Stanislav Halaida.[33]