Bella-Rae Birch died in hospital from her injuries after emergency services were called to her home in the Blackbrook area of St Helens at 3.50pm on Monday, Merseyside Police said.
Neighbours described trying to save the “beautiful” toddler after hearing her parents’ screams.
In a statement, her family thanked the community for their support.
They said: “We would ask that we are now allowed some space and time to try and come to terms with the tragic loss of our much-loved Bella-Rae.
“She will be sadly missed but never forgotten.”
Superintendent Steve Brizell said: “Bella-Rae has lost her life in the most unimaginably terrible circumstances and our thoughts are first and foremost with the family and the wider community at this devastating time.
“Our specialist family liaison officers are providing the family of Bella-Rae with support and officers will remain in the area in the coming days to provide further reassurance and support for the wider community.
“At this stage we understand that the family dog they had bought just a week earlier has attacked Bella-Rae inside the family home.”
The force said the dog had been humanely destroyed and work was ongoing to determine its breed and its previous owners.
One neighbour, who gave her name as Jordan, said she ran to help Bella-Rae.
She said: “We pulled up from school and heard screaming. I just ran over to try to help and started CPR until the paramedics took over.”
The neighbour, who is due to go to university to train as a paramedic, said the child’s parents were “hysterical”.
She added: “I didn’t see the dog, I was just focusing on helping the baby.”
Another neighbour who helped to give CPR said: “The parents doted on the children. It’s too upsetting to say anything else.”
Neighbour Joanne Matthews, 53, said: “She was such a beautiful little girl, toddling about.
“I’d see the family in passing, just to say hello, and they were always very pleasant.”
Ms Matthews said she had seen the ambulance outside the house on Bidston Avenue at about 3.50pm on Monday and then about 10 police cars, including a van, arrive.
She said: “I saw them bring the dog out. I couldn’t tell what breed it was but from the back it looked like a Staffordshire bull terrier or pitbull.”
Another neighbour, who did not want to be named, said she came home from the shops at about 4.30pm and the area was filled with police.
She said: “My friend gave me a lift and I said to her there’s got to have been a murder.
“The mum was on the field at the front of the house crying, she was hysterical.”
On Tuesday morning police officers were stationed outside the house, which was cordoned off with police tape, and a forensics officer was seen going inside the property.
Flowers were laid outside the house by friends and neighbours.
Anyone with information can contact Merseyside Police on Twitter @MerPolCC, or Merseyside Police Contact Centre on Facebook, with reference number 22000196837.
They can also contact Crimestoppers anonymously on 0800 555 111.