Little Olivia Pooley was admitted to Colchester General Hospital to have treatment after a hair became tightly wrapped around her toe.
Staff gave her morphine as they removed the 'hair tourniquet' from her left foot. Medics then allegedly admitted to her dad, Daniel, that they had given her too much of the drug.
Daniel, 27, said Olivia underwent the treatment after the piece of hair wound itself around her toe and caused her 'cheese wire effect' pain.
He told the BBC that after the hair had been removed, his little girl's toe had been left 'in a bad way' and she was in a lot of pain and given morphine.
"Forty five minutes afterwards we were advised she had been given twice as much as she should have been given, 4mg instead of 2mg," Mr Pooley said.
Olivia then reportedly became 'extremely drowsy and very difficult to wake'.
"It slowed her breathing down and she was moved to resuscitation for constant monitoring," her dad said.
Olivia has now made a full recovery, but Colchester Hospital University NHS trust said that they will be speaking to the family to include them in an inquiry into the events.
Sue Barnett, deputy chief executive at Colchester Hospital University NHS Foundation Trust, said: "We will be contacting Mr and Mrs Pooley by the end of next week, asking them to take part in that investigation so that we fully understand the issues involved."