Head of paediatric surgery Dr Joe Crameri, who is leading the delicate procedure, is due to provide details of the surgery this morning.
The girls are joined at the torso and share a liver and possibly a bowel, their doctor said, adding "our challenge will be to reconstruct their abdominal walls to close it over", the BBC reported.
They were joined at the torso and their livers were joined.
The worst-case scenario would have been if the girls shared a component that was vital to both. The girls are expected to remain in hospital for at least a week, Crameri said.
A pair of Bhutanese conjoined twins have been successfully separated after a marathon surgery involving 18 doctors.
"We didn't find surprises", said lead pediatric surgeon Joe Crameri, "we knew the liver would be connected.it was divided successfully without any major bleeding".
Get well soon Nima and Dawa!
"There will be challenges over the next 24 to 48 hours as with any surgery, and we feel quietly confident that we will have a good result", he said.
The family was brought to Australia from Bhutan by Children First Foundation, an Australian-based charity. From that point on, to avoid confusion, Nima was known as "Green" and Dawa as "Red".
Their mother was feeling "a little bit scared" about the procedure, but had shown "extraordinary calmness" so far, the BBC quoted Elizabeth Lodge from the charity as saying.
"She still has this extraordinary calmness about her, which is just unbelievable".
A surgery to separate the conjoined twins Nima and Dawa will take place at the Royal Children's Hospital (RCH) in Melbourne, Australia today.
One of the biggest operating theatres was commandeered for the procedure, which involved two teams of anaesthetists - one for each sister.
Several members of the surgical team had previously worked on the successful operation to separate conjoined Bangladeshi twins Trishna and Krishna in 2009.
Dr Karma Sherub, a pediatric surgeon with JDWNRH, who is now in Melbourne said that the twins look good and have become strong and healthy. They could stand but only at the same time.