Najiyah died from nitrous oxide poisoning and because her brain was starved of oxygen.
Najiyah Hussain was taken to Newham General Hospital after suffering a fit at her home in East London.
She died after being given nitrous oxide instead of oxygen.
Nitrous oxide, or laughing gas, is used as an anaesthetic.
Najiyah's father Akmul Hussain said from his home in Manor Park, east London: "She should have had the right treatment.
"She was a very lovely girl ... a lovely girl and she had just started nursery.
"I feel terrible. How could they do that? It should not happen to anybody else."
"They hope that her death causes hospitals everywhere to double-check their procedures to ensure that something like this cannot happen to somebody else."
Najiyah arrived at the hospital's accident and emergency department on 18 January. She had gone into convulsions after receiving a flu jab at her GP practice.
She was given a mask to help her breathe.
However, instead of being connected to a supply of oxygen, the mask fed her nitrous oxide.
Internal investigation
Najiyah died from nitrous oxide poisoning and because her brain was starved of oxygen.
Najiyah's sister Salema Khatun said: "All the other times she has gone into hospital she has been in good hands, and she has come home - but she hasn't this time.
"It is hard to understand, it such a silly mistake. Oxygen and nitrous oxide are completely different things.
"Oxygen is such a simple thing to give, and yet they still made that mistake."
The Trust has begun an internal investigation, and a doctor has been suspended.
A spokesman for the hospital said senior representatives for the Trust have met the parents and expressed their sincere condolences.
Dr Charles Gutteridge, medical director of the hospital, said: "We were concerned about some of the events around that day and so have set up an internal inquiry.
"The panel consists of some staff within the hospital and an independent external expert."
Modern equipment
Experts say that modern equipment should make it impossible for such a mistake to occur.
Mike Stone, director of the Patients' Association, said: "It is tragic, harrowing case. There should be measures that stop things like this happening."
Mr Stone said different gases should be colour coded and fitted with nozzles that can only be used by specific cylinders.
He said doctors and nurses were under constant strain, and that action had to be taken to relieve pressure on them.
But he said training should ensure that such tragic mistakes could not happen.
The tragedy follows two other high profile deaths blamed on NHS errors in recent weeks.
Wayne Jowett, 18, died at Queen's Medical Centre, Nottingham, after a cancer drug was injected into his spine, rather than into a vein.
And a patient died at the Royal Sussex County Hospital in Brighton after local anaesthetic was injected into a vein rather than his spine.
An inquest into Najiyah's death was opened and adjourned at Walthamstow Coroner's Court on Wednesday.
A file on her case has been sent to the Crown Prosecution Service.