The former Superior General of Missionaries of Charity was suffering from various ailments.
She took over the running of the charity after Mother Teresa died in 1997 and ran it for 12 years before Sister Mary Prema took over in 2009.
The Missionaries of Charity cares for the homeless and dying in Kolkata.
Archbishop of Kolkata Thomas D'Souza told the AFP news agency that Sister Nirmala "passed away peacefully, surrounded by sisters praying for her".
"She had been suffering from a heart disease for quite some time, but she never lost her smile and was always cheerful."
He said Sister Nirmala "carried forward Mother Teresa's legacy of compassion, gentleness, service to the poorest of the poor and holiness of life".
"Even after she ceased to be head of Missionaries of Charity, she served the people with the same zeal as she did before."
Reports say Sister Nirmala's body will be kept at the Missionaries of Charity headquarters in Kolkata and her funeral will be held on Wednesday afternoon.
Prime Minister Narendra Modi led the tributes to the deceased nun.
West Bengal Chief Minister Mamata Banerjee said she was "saddened" by Sister Nirmala's death.
The Missionaries of Charity has about 4,500 nuns and hundreds of homes around the world.

Strikes planned for Christmas were suspended after the new offer was made but members rejected it by 7-1.
The strike affects "mixed fleet" staff who joined after 2010 and account for 15% of the total cabin crew.
BA said it planned to "ensure that all our customers travel to their destinations".
It said further details for customers would be published on Friday 6 January, once it had finalised contingency plans.
"We are extremely disappointed that Unite has once again chosen to target our customers," the company said.
"We are now focused on protecting our customers from this unnecessary and completely unjustified action."
The dispute concerns about 4,000 staff who have joined the airline since 2010 on "mixed fleet" contracts, who do not feel they are paid enough. About 2,500 of them are Unite members.
Earnings were advertised between £21,000 and £25,000 but, in reality, start at just over £12,000 plus £3 an hour flying pay, Unite said.
The union said its members rejected the new offer by 7-1.
"British Airways is needlessly provoking strike action by refusing to extend the mandate of the strike ballot and allow meaningful talks to take place," Unite's national office Oliver Richardson said.
"Instead of listening to why its mixed fleet cabin crew rejected the offer negotiated at Acas, British Airways has sought instead to try and bully a workforce of young men and women who are trying to eke out a living on poverty pay."
However, he added that the union was "hopeful that a negotiated settlement which meets our members' aspirations can be achieved" and urged BA to "engage constructively".

Charlie Dunn, from Tamworth, died in hospital after being found submerged in the Blue Lagoon area at Bosworth Water Park in Leicestershire on 23 July.
Lynsey Dunn, 28, and Paul Smith, 36, who are believed to be the boy's mother and stepfather, have been charged with manslaughter by gross negligence.
They are also accused of seven child neglect offences.
These offences are alleged to have taken place between April 2012 and July 2016.
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In addition, Mr Smith has been charged with two counts of witness intimidation alleged to have taken place in March 2016 and June 2017; one count of threats to damage or destroy property by fire in June 2017; and eight driving offences which are alleged to have taken place in January and June 2017.
Both defendants, of Caledonian, Glascote Heath, Tamworth, appeared at Leicester Magistrates' Court earlier and are due to appear at Leicester Crown Court on 5 July.
Sister Nirmala, the nun who succeeded Mother Teresa as the head of a charity in the Indian city of Kolkata (Calcutta), has died, aged 81.
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British Airways cabin crew will stage a 48-hour strike from 10 January after rejecting a new pay offer, the Unite union says.
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A man and woman have been charged with the manslaughter of a five-year-old boy who drowned in a lake.