Mohammad Sudais, who lost his older brother and parents in the gas explosion at their home in Peshawar leaving him with 80% full thickness loss to his face, was brought to Glasgow's Royal Hospital for Sick Children for treatment.
He has also been visited by Health Secretary Alex Neil who gave permission to allow him to fly to Scotland.
The baby was successfully operated on last week and is now able to close his eyes as well as smile. Doctors are preparing to carry out more reconstructive surgery in the coming weeks.
Mohammad Asif, the baby's uncle, has lived in Glasgow for 15 years and plans to adopt the infant. He described the Health Secretary as "a great man" following the visit to his nephew.
Robina Qureshi, chief executive of the charity Positive Action in Housing, has been supporting the family. She said Mr Neil's actions had saved the baby's life and he showed "humility and compassion" when he visited Mohammad.
The baby travelled to Scotland on an Afghan passport with a six-month medical visa, although it is hoped he will be granted permission to stay in Britain permanently by the UK Borders Agency.