Christopher Hannah was driving a car which mounted the pavement and struck Sophie Brannan in Sandbank Street, Maryhill, on 14 November last year.
Her friend Chelsea Sommerville, and Chelsea's uncle, Joseph Lloyd, were also badly injured.
At the High Court in Glasgow, 33-year-old Hannah, who is said to be a heroin addict, admitted a charge of culpable homicide. Sentence was deferred.
Following his guilty plea, the investigating officer Det Insp Colin Hailstones said: "This was an appallingly tragic case.
"Many people in the community in Maryhill were deeply affected by this incident, particularly as it resulted in the death of young Sophie Brannan, whose life was sadly taken away from her at the age of 11 and at the hands of someone under the influence of drugs.
"Police officers worked tirelessly to piece together the events of that fateful night and I must thank the many members of the local community and those injured who helped provide information about the incident."
He added: "The strength shown from the community towards Sophie's family and the other people who were injured is unquestionable and much support continues to be given to them."
The court heard that Hannah was driving a hired Vauxhall Astra in the area and lost control.
He then fully mounted the pavement before hitting the gable end of a nearby building.
The car continued to career forward eventually ploughing into Sophie and the two people with her from behind.
Prosecutor Allan Nicol said: "This caused them to be thrown onto the bonnet, windscreen and roof of the vehicle before falling onto the ground."
Another driver witnessed the crash and turned back to help Sophie.
More witnesses saw Hannah speed onto nearby Maryhill Road with his badly-dented car being "driven erratically".
He then crashed into a taxi whose driver decided to follow Hannah, who eventually stopped a short distance away.
Mr Nicol added: "At that point, the accused was trying to pull the damaged front bumper then he stopped and walked away before starting to run off.
"Both the taxi driver and his passengers noticed that he was under the influence of some substance."
Medics found Sophie lying on the pavement. She was rushed to the Royal Hospital for Sick Children at Yorkhill where she was found to have swelling to her brain, several fractures and broken bones.
The schoolgirl remained in intensive care overnight but was pronounced dead on the morning of 15 November.
The court heard that Chelsea suffered a serious leg break which will require long-term physiotherapy.
She continues to wear a cast several months after the tragedy.
Mr Nicol added: "She also suffers episodes of withdrawal and angry outbursts.
"She is due to attend for psychiatric assessment in order to come to terms with her injuries and the loss of Sophie."
Chelsea's uncle also suffered a number of fractures, but was initially released from hospital two days later.
However, he eventually required surgery and it is likely he will have "long-term restriction" of movement in his right shoulder.
Police discovered the Astra had been hired in Hannah's name on 14 October.
The court also heard that shortly after the incident, police searched the home of Hannah's partner, where he had been staying.
Mr Nicol said tin foil wraps with signs of drug use were discovered.
The prosecutor told the court: "On the day in question, she suspected he had been taking heroin as she was aware he is heroin-dependent."
His partner revealed Hannah had come home earlier that night "in shock".
Hannah had claimed he had hit someone, but did not know who.
The killer said he was going to a house in the city's south side, but would return.
Police were at the house when Hannah then called his girlfriend. She handed the phone to an officer and Hannah said: "I'm sorry - I didn't mean to do it.
"It was a total accident. I lost control of the car and I panicked. I'm going to hand myself in tonight."
The court heard that in the early hours of 17 November, Hannah turned up at a house in the city's Mount Florida area, but those inside decided to call police.
Officers turned up and arrested Hannah, who initially tried to escape. He was later searched and eight grams of heroin were discovered inside his boxer shorts.
The court heard how the occupants of the Mount Florida property later told police Hannah had called them on the night of the killing.
Hannah said at the time: "There's been an accident - watch the news. The police have fitted me up.
"I tried to get away and I have hit something. I don't know what."
Mr Nicol told the court: "He went on to say he had driven like a madman and that the something he had hit could have been a wheelie bin...but he didn't hang about to find out."
Hannah asked to stay for a while as the police were after him "because he had killed a wee lassie".
Accident investigators later concluded Hannah was solely to blame for what happened.
It was stated he took "a conscious decision" to drive while "impaired".
They added that the reason for him not stopping was because he knew he had taken drugs and was "aware of the ramifications".
As well as culpable homicide, Hannah also pleaded guilty to dangerous driving, attempting to defeat the ends of justice and possessing heroin.
It emerged he already had 14 previous convictions for crimes including having an offensive weapon and road traffic charges.
He was on bail at the time, having been freed from Glasgow's Justice of the Peace Court two months before the crash.
Hannah's advocate Thomas Ross said: "He has asked to state publicly his apologies for those affected by this terrible tragedy."
Lord Bannatyne remanded Hannah in custody and deferred sentencing until May.
A man has admitted killing an 11-year-old girl in a hit-and-run in Glasgow.