Iranian legend Yazdani proved too good for Punia in the 86kg category final.
India have won five bronze medals so far in wrestling in this edition.
Antim saved the day for the Indian contingent which badly needed a medal after Narinder Cheema (Greco-Roman 97kg), Naveen (Greco Roman 130kg) and Pooja Gehlot (women's 50kg) were ousted.
The other Indians in contention -- Gyanender (60kg), Neeraj (67kg) and Vikas (77kg) -- could not win a single round.
Delhi Police cited alleged incidents from Tajikistan in its case against Singh, claiming that these events show his awareness of his actions.
The high-scoring bout ended with a technical superiority win for the Indian, who became the first Indian wrestler -- male or female -- to lock a quota for the next year's Games.
She later outplayed Poland's Roksana Marta Zasina by technical superiority to move into the quarterfinals.
Vinesh's outburst comes after the United World Wrestling provisionally suspended the Wrestling Federation of India for not conducting elections on time.
The Indian wrestlers will have to compete as 'neutral athletes' at the Olympic-qualifying World Championships, starting September 16.
Indian women's team won the team title at the world championship.