Four-year-old Violet Grace Youens died on Saturday, a day after the vehicle mounted the pavement in St Helens.
The girl's 55-year-old grandmother, Angela French, was also hit by the car and remains in a serious but stable condition in hospital.
The man held, 27 and of St Helens, is also detained on suspicion of causing serious injury by dangerous driving.
Another man, aged 23, is believed to have left the country, said Merseyside Police.
"He left approximately three hours after the incident," said Det Ch Insp Chris Sephton.
"We have been in contact with the authorities of the country where we believe he is... We are doing everything we can to try and identify where he is now and get him back over here."
A third man, aged 23 and of Prescot, was arrested on suspicion of being a passenger in a stolen vehicle.
He has been released pending further inquiries, police said.
The crash in Prescot Road involved a black Ford Fiesta, which then crashed into a lamppost.
Police issued CCTV images of two men, aged about 15-20, they want to speak to who were seen running away from the area.
Violet's mother Rebecca agreed to donate her daughter's pancreas and kidneys following her death, police said.
She posted a tribute to her daughter on Facebook, saying "my brave baby girl saved two lives by donating her kidneys and pancreas. I am truly heartbroken but proud of my little fighter!"
An online page raising money for Violet's family has so far reached Â£9,000.
A man has been held on suspicion of causing death by dangerous driving after a girl was struck by a car.