The miracle involved the healing of a Brazilian man with several brain tumours in 2008, the Vatican said.
Mother Teresa died in 1997 and was beatified - the first step towards sainthood - in 2003.
She won the Nobel Peace Prize for her work with the poor in the slums of the Indian city of Kolkata (Calcutta).
"The Holy Father has authorised the Congregation for the Causes of Saints to proclaim the decree concerning the miracle attributed to the intercession of blessed Mother Teresa," the Vatican said on Friday.
She is expected to be canonised in Rome in September.
Sister Christie, a spokesperson for the Missionaries of Charity Mother Teresa founded in 1950, told the BBC that they were delighted by the news.
"Obviously all of us at the Missionaries of Charity are extremely happy. But we do not have any plans to celebrate this announcement as yet," she said.
The Missionaries of Charity now has more than 4,500 nuns worldwide and is headquartered at the Mother House in central Kolkata.
In the city, the charity runs 19 homes - for women, orphans and the aged - where thousands of destitute people live. It also runs a school for street children, an Aids hospice and a leper colony.
Though the order's "service to the poorest of poor" has generated much appreciation worldwide, it's not without controversies.
The most recent controversy involves shutting down of its adoption centres in India.
The charity said it was forced to close the centres because India's new adoption laws, allowing single, divorced and separated couples to adopt, went against its religious views.
Mother Teresa was born Agnes Gonxha Bojaxhiu in Skopje, now Macedonia, then the Ottoman Empire, in 1910. Her family were ethnic Albanians, and devout Catholics.
She dedicated her life to caring for impoverished and sick people in Kolkata.
Known as the "saint of the gutter", she earned worldwide acclaim for her efforts.
Her critics, however, accused her of peddling a hardline Catholicism, mixing with dictators and accepting funds from them for her charity.
Her supporters justified the funding, saying it did not matter where the money came from as long as it was used to help the poor.