Mohammad Sudais was moved to the high dependency unit at the Royal Hospital for Sick Children, Yorkhill, Glasgow, after a team led by consultant Stuart Watson performed the procedure on his eyes and mouth.
The infant, who was found whimpering underneath the remains of his plastic carry cot, was flown to Scotland a week ago by his uncle Mohammad Asif, who has lived in Glasgow for 15 years, following the intervention of the Scottish Government and a public appeal which has so far raised £15,000.
The baby, who has suffered 80% full skin loss to his face, lost his father, mother and 13-month-old brother in the weeks following the blast at his home in Peshawar, on December 16, which is believed to have been caused by a gas outage.
Robina Qureshi, director of the Positive Action for Housing, assisted Mr Asif with his campaign to get his four-month-old nephew, who he plans to adopt, flown to Scotland for treatment.
Ms Qureshi said last night she was "so relieved" after receiving news the operation - likely to be the first of many over several years - had been deemed a success.
She said: "Over 500 people have donated from all over Scotland and further afield, it's been overwhelming.
"Everyone had to ensure the child was going to be transferred safely and there were a lot of logistics involved in bringing him over from what is a very insecure place. The Scottish Government supported it and [Health Secretary] Alex Neil has to be thanked - he said 'yes, he can be treated in a Scottish hospital' and pulled out all the stops."
Mohammad survived a scare on Tuesday night when he stopped breathing and had to be resuscitated. Ms Qureshi said the baby would not have survived had he still been in Pakistan.
Money is still being raised and will be used to fund treatment for Mohammad as he gets older.
"You can see overwhelmingly that there's a baby who needs a lot of love," Ms Qureshi added. "He's adorable. This child is clearly a fighter and will one day do this country proud."
The baby has a six-month medical visa, but it is hoped a permanent visa will soon be granted by the UK Borders Agency. It is planned that once he is well enough to leave hospital, he will live with Mr Asif, a language consultant, his wife Naseema and their four children at their Carmyle home.
Though his face was badly burned, Mohammad's body is unmarked as he was wrapped in blankets when the explosion occurred.
To donate to the appeal, visit www.justgiving.com/muhammadsudaisburnstreatmentglasgow