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."
A driver has been convicted of a hit-and-run crash in Leeds which left a girl permanently disabled and her brother seriously injured.