Aradhana and Stuti Yadav, 11-month-old girls with fused livers and two hearts joined by a common membrane, had been dumped in a hospital hours after birth.
A specialist team of 23 doctors and 11 nurses from India and Australia took part in the operation, which was paid for by public donations after the girls' plight was featured on national TV.
The outpouring of generosity, with thousands of small donations by members of the public, is unusual in India where girls are often not valued in rural communities and disabled babies are often abandoned.
Public generosity was matched by a £22,000 fund from the Madhya Pradesh state government.
The girls have been cared for by hospital staff since their birth, with three nurses taking shifts to act as surrogate mothers. Surgeons at the under-equipped Paddar Hospital spent months appealing for international help and preparing for an operation.
Born into a poor family in the small town of Betul, the girls' father is said to be a farmer with a smallholding of land. Hospital staff tracked the family down and agreed to look after the children if the parents developed a relationship with the girls.
"They are very poor and even the delivery had been done under a government scheme for healthy deliveries. After the babies were born it was clear they would not have been able to provide medical attention needed for them. So, the hospital decided to look after them in the near future till it became clear that surgery would be possible," said a spokesman.
Hospital authorities said they were pleased the parents wanted to take their girls home after surgery, describing it as the best possible outcome.
"We are happy that they will finally get to be a family and both the girls will get a proper home," said Vikas Sonwani, assistant administrator of Paddar Hospital.
The parents, Hari Ram Yadav and Maya Yadav, are both believed to be in the late twenties and already have a six-year-old son. They were not available for comment.
The day long operation to separate Aradhana and Stuti began at 8am on June 20.
"They had separate hearts and livers but the problem was both the hearts were in single sac," said Mr Sonwani.
"All the organs in our body are covered with a dedicated sac. In case of the heart, its called pericardium. But in these babies, there was only one common pericardium. Similarly they had separate but fused livers," he added.
During the operation a team of surgeons separated the girls' hearts and transplanted the organs into each chest cavity. They then separated their livers.
Both the twins have been kept on ventilators and they will be under close medical observation for about 48 hours.
Doctors are optimistic both girls will make a full recovery.
Check out these wonderful pictures of previously conjoined twins seeing each other's faces for the first time: