The award was announced on Tuesday by bestselling Booker Prize-longlisted author Max Porter in his role as chair of the five-member voting panel, at a ceremony at London's Tate Modern.
It is the first time the award has been given to a collection of short stories.
Bhasthi is the first Indian translator - and ninth female translator - to win the prize since it took on its current form in 2016.
Mushtaq is the sixth female author to be awarded the prize since then.
Written in Kannada, which is spoken by about 65 million people, primarily in southern India, Porter praised the "radical" nature of the translation, adding that "It's been a joy" to listen to the evolving appreciation of the stories by members of the jury.
"These beautiful, busy, life-affirming stories rise from Kannada, interspersed with the extraordinary socio-political richness of other languages and dialects," Porter said.
"It speaks of women's lives, reproductive rights, faith, caste, power and oppression."
The book, which beat five other finalists, comprises stories written from 1990 to 2023.
They were selected and curated by Bhasthi, who was keen to preserve the multilingual nature of southern India in her translation.
Mushtaq, who is a lawyer and activist as well as writer, told a short list reading event on Sunday that the stories "are about women - how religion, society and politics demand unquestioning obedience from them, and in doing so, inflict inhumane cruelty upon them, turning them into mere subordinates".
The Stg50,000 ($A104,000) prize money is to be divided equally between author and translator.
Each is presented with a trophy, too.
The International Booker Prize is awarded every year.
It is run alongside the Booker Prize for English-language fiction, which will be handed out later in the year.
