Competing in the 56kg category, the 15-year-old Sonam defeated Japanese Sena Nagamoto 3-1 in the final.
She took a 2-0 lead in the first round and then conceded a point in the second round before adding one more towards the end of the bout.
Neelam, competing in the 43kg category, defeated Roxana Alexandra of Romania 6-4 in the bronze medal bout.