Verma beat Mete Gazoz of Turkey 6-4 (29-28, 30-29, 27-27, 29-30, 29-29) in a closely-fought bronze medal match.
Earlier in the day, Verma had beaten Hiroki Muto of Japan 6-4 in the quarterfinals before losing to Marcus D'Almeida of Brazil 4-6 in the semifinals.
This is the second medal for India in the Games after weightlifter Ragala Venkat Rahul bagged a silver Thursday last.
In wrestling, Reshma Anil Mane finished fifth in women's freestyle 60kg.