Sabah Saleem, 11, and three-year-old Rayhan were hit by a car driven by 37-year-old Mohammed Khalil Anwar on Brudenell Road in August 2012.
Leeds Crown Court heard that Anwar, of Brudenell Mount, failed to stop as he feared being attacked by local people.
Anwar is due to be sentenced on 29 November.
Sabah and Rayhan, who were 10 and two at the time of the incident, were knocked down by Anwar's Vauxhall Astra car on 18 August last year.
They were returning from a greengrocer's shop with food for the family's Eid celebrations which were due to take place the following day.
Both underwent surgery at Leeds General Infirmary.
While Rayhan was discharged from hospital a few days later, Sabah remained in hospital for about six months and surgeons operated on her several times.
The court was told she had been left with "significant and permanent" disabilities as a result of the crash.
During his trial, Anwar told the court he was in control of his car at the time of the collision, but he had no time to react when Sabah and Rayhan ran out in front of him.
The court heard Anwar was driving at nearly 35 mph along Brudenell Road when he collided with the two children, about 15 mph faster than the speed limit.
Anwar admitted he did not know the speed limit was 20 mph along that stretch of road, even though he had regularly driven along there over the past 12 years.
He told the court he drove off after hitting the children as he was afraid he would be attacked.
Following the verdict, Sabah said she still could not remember anything about the accident.
While she still found it "very difficult" to walk, she was now feeling "okay", she said.
Sabah's father, Saleem Rafique, said his "very brave" daughter was still recovering.
The 11-year-old continues to suffer from double vision, balance problems and memory loss, he said.
"She was a really bright kid, doing really well at school and very determined to succeed.
"Unfortunately, this happened and that has put her back.
"She's tough, she's a real fighter but we don't know whether she will be the same as she was before. It's very hard to tell."

Beach Buddies, established in 2006, co-ordinates weekly operations around the Isle of Man's coast.
A spokesman said the cash will mean the organisation can "expand on its work" which includes launching an educational programme in the coming year.
The government awarded the grant and said its work had "encouraged more people to use the outdoors".
The island's environment department said Beach Buddies, which has about 7,000 volunteers, had complemented several government strategies which also included growing visitor numbers.
Beach Buddies organiser Bill Dale said: "We now believe they [the beaches] are the tidiest in Europe."
New initiatives also include more bins at various beauty spots around the island.
It also aims to establish an online litter hotline where the public can report problems.

Nicole Clark used her mother Avril Stewart's bank details to set up online accounts.
The 30-year-old, from Woodside, admitted stealing the money between 8 August and 26 August last year.
Depute fiscal Robbie Brown told Perth Sheriff Court that the accused's mother had not recouped any of the stolen money.
Sheriff William Wood deferred sentence for the preparation of social work reports.
A driver has been convicted of a hit-and-run crash in Leeds which left a girl permanently disabled and her brother seriously injured.
[NEXT_CONCEPT]
A beach clean-up charity has been given a £30,000 grant to help keep Manx beaches the "tidiest in Europe".
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A Perthshire woman lost £2,020 she had stolen from her mother gambling at an online casino, a court was told.