Nine-year-old Olivia died after the shooting on Monday night when a 35-year-old man, unknown to the family, ran into the terraced house in Kingsheath Avenue, in the Dovecot area of Liverpool.
It came as he attempted to get away from a shooter, Merseyside Police said.
And the Prime Minister has shared his grief at the tragedy that has shocked the nation.
Mr Johnson wrote in a statement: "My thoughts are with Olivia Pratt-Korbel’s family and the people of Liverpool following this horrific, senseless shooting.
"This is an unimaginable tragedy and we will ensure Merseyside Police get whatever they need to catch those responsible and secure justice for Olivia."
Prime Minister Boris Johnson
Peter Byrne
Olivia Pratt-Korbel died on Monday evening
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Olivia's mum Cheryl Korbel, 46, was shot in the wrist as she tried to close the door on the gunman while her nine-year-old daughter stood behind her.
The man who had entered their home suffered gunshot wounds to his upper body and, as Olivia lay dying, was picked up and taken to hospital by friends driving a dark-coloured Audi.
Speaking at a press conference on Tuesday, Merseyside Police Chief Constable Serena Kennedy said: “We will not rest until those who are responsible are put behind bars. I can guarantee that no stone will be left unturned.”
Detective Chief Superintendent Mark Kameen said Olivia had been at home with her mum and two older siblings when Ms Korbel opened the door after hearing gunshots fired outside.
He said a figure, wearing a black padded jacket, a black balaclava with a peak, dark trousers and black gloves, fired shots at two men walking along Kingsheath Avenue, causing both of them to run away.
He continued: “As that was taking place the person with the gun has followed the male to this property.
“That person has also tried to force entry to the property and has managed, it would appear, to put their hand through the open door as Cheryl continued to try and close it.
“A shot has been fired which we believe has hit Cheryl, injuring her and then fatally wounding Olivia. Olivia at that time, we believe, was stood directly behind Cheryl.”
Olivia was taken to Alder Hey Children’s Hospital but later died.
The Audi used to transport the injured man has since been seized by police.
Mr Kameen urged the person responsible to come forward.
