Galustyan debuted on the ISU Junior Grand Prix (JGP) series, finishing 7th in Košice, Slovakia in September 2013, and 10th in Ostrava, Czech Republic the next month. She made her senior international debut in December, placing 5th at the Ukrainian Open. In March 2014, she competed at her first ISU Championship, the World Junior Championships in Sofia, Bulgaria, but was eliminated after placing 27th in the short program.