The boy was given an overdose of anaesthetic during the operation at Leicester Royal Infirmary in 2011.
His mother said her son's life was "turned upside down" by what happened.
Liability was admitted by University Hospitals of Leicester NHS Trust and an apology was made on its behalf at the High Court in London.
The Leicestershire teenager, who cannot be identified, suffers from severe physical and mental difficulties as a result of the injury.
He also needs a wheelchair and is unable to feed himself independently.
An internal investigation by the trust revealed there was a failure to flush a tube used during the surgery with saline solution.
This resulted in an overdose of anaesthetic causing him to suffer a cardio-respiratory arrest, cutting off oxygen to the brain.
He had to be resuscitated but there was a delay before an attempt was made, the family's lawyers said.
The teenager was transferred to intensive care and given artificial breathing support and ventilation.
He was then put into an induced coma for several days.
His mother said: "My son's life and ours have been turned upside down impacting heavily on him and on us as a family."
She said the money will support his needs for the rest of his life and added that the case was a chance to highlight the error so "lessons can be learned".
Mr Justice Goss said the financial settlement, which remains confidential, was "appropriate and just" and would provide financial security for the family and allow the teenager to have the "best possible life".
A teenager who suffered a severe brain injury during an operation to remove his appendix is set to receive a seven-figure sum in damages from the NHS.