Danielle McLaughlin's body was discovered in a secluded spot in Canacona, a popular area for holidaymakers in the south of the Indian state.
The 28-year-old is believed to have been visiting the area with a friend to celebrate the Indian festival of Holi.
Local media reported the former Liverpool John Moores University student was discovered unclothed and she had marks on her head and face.
One man who has been arrested has confessed to her rape and murder, according to police.
Andrea Brannigan, the victim's mother, said: "The family would like to express our thanks to all who have got in touch since receiving this awful news.
"As you can expect we are finding it very difficult at this trying time.
"We want to thank the Irish and British consulates, along with Colin from the Kevin Bell Repatriation Trust, and many friends who have assisted at this time.
"Danielle will be sadly missed by us all."
Ms McLaughlin was the eldest of her five daughters.
Her body was found in a remote area near Deobagh Beach on Tuesday morning, according to reports.
Ms McLaughlin grew up in Buncrana, Co Donegal, and moved to the UK where she studied and worked in Liverpool before travelling for a time in Australia.
She had returned to her home town before setting off to India on a backpacking adventure last month.
On her own Facebook page, dated February 22, Ms McLaughlin posted: "Thank you to all my friends and family for making home so special and always looking after me. I am very grateful and the luckiest person I know... Off on another adventure... "
Dozens of people posted tributes on the social media website.
One said: "Sleep peacefully Danielle, my thoughts are with your family and loved ones. The horrors you have suffered are over x x x"
Another wrote: "Love you Danielle, you had the most beautiful soul."
It is understood a post-mortem examination has taken place and her body will be released on Thursday.
A family friend is expected to accompany the body home to Ireland.
Christy Duffy, a close friend, has already raised several thousand euro on a fundraising website to help pay for the funeral and repatriation costs.
"She was beautiful, kind-hearted and funny. She just loved life. We are devastated," he said.
"They say the brightest stars burn the quickest and no other analogy could describe Danielle's life better."
Rosalee Grant, principal at Scoil Mhuire, where Ms McLaughlin went to school, described her past pupil as a "lovely, happy go lucky student".
"She was a very caring individual. She took part in school musicals, she loved Irish dancing, she also liked athletics and soccer," she said.
"She was an all rounder. Happy, carefree and caring. She went off to college in Liverpool and then went travelling.
"This is a devastating time for her family and the whole town. It's a big shock for the staff and pupils.
"The staff remember her fondly. She was a student we were very proud of. It is very sad, she was in the prime of her life."
The Department of Foreign Affairs in Dublin said Ms McLaughlin was travelling to India under a British passport.
"It has been confirmed that the woman travelled to India using a British passport, so the local British consulate have the lead responsibility in terms of dealings with the Indian authorities," a spokesman said.
"The Irish Embassy is liaising closely with the British authorities and an Irish consular official is travelling from New Delhi to Goa."