Susan Allan died after being struck by Richard Quaye's car when he pressed the accelerator instead of the brake during a "momentary lapse of concentration."
Mrs Allan, 66, died at the scene of the crash on Long Row, Horsforth, on December 20 last year.
A court heard the pensioner had been looking forward to spending her first Christmas with her granddaughter and her family had been left devastated by the her loss.
Social worker Quaye, 44, had his two daughters, aged six and eight, in the car at the time of the incident at 2.30pm.
Leeds Crown Court heard Quaye had worked a 12-hour shift the previous night and only slept for two hours prior to the incident.
He was jailed for 16 months after pleading guilty to causing death by dangerous driving.
Mark Kellett, prosecuting, said Mrs Allan, of Park Grove, Horsforth, had been pushing her granddaughter in a pram and had the family dog with her.
CCTV footage of the incident showed Quaye's vehicle failing to turn as the road curved to the left.
The VW Tiguan crossed into the opposite carriageway before mounting the pavement and hitting the pensioner.
The vehicle struck a wall before stopping. The pram was also hit by the car but the baby was not injured.
Mr Kellett said: "The defendant got out to check the baby in the pram.
"He was wandering around like he did not know what to do with himself and asking if she was OK."
Emergency services were called but Mrs Allan died at the scene from multiple injuries.
Quaye was arrested and told police he had not been distracted by anything in the car.
He told officers that when he realised he had gone into the opposite carriageway he tried to brake but instead put his foot on the accelerator.
Quaye, of Holtdale Place, Adel, said he had worked a 12-hour shift, finishing at 7.15am that morning, and had then slept for two hours.
The prosecutor said Mrs Allen's husband Michael had provided a victim statement describing how the mum-of-two was looking forward to Christmas with their first grandchild.
He said Mr Allan had also described how he had not only lost his wife of 43 years but had also lost his best friend.
Andrew Epsley, mitigating, said Quaye was a hard-working, loving, family man who had no previous convictions.
He described how Quaye had moved to the UK from Ghana in 2001 to study at university in Leeds.
He had worked for Lloyds bank, the NHS and at a secure children's centre working with young offenders before pursuing a career as a social worker in 2012.
Mr Epsley said Quaye was a committed Christian who did work on behalf of his church and for charities.
The barrister said Quaye had written a letter to Mrs Allan's family expressing his remorse.
Mr Epsley said: "Nothing I say should be taken as intended to minimise the loss suffered by the Allan family."
Jailing Quaye, The Recorder of Leeds, Judge Peter Collier, QC, said: "It may be that there was simply some momentary lapse of concentration and you were not able to get the car on to the road.
The judge added: "It may be that part of the inattention was caused by the fact that you had been working over night.
"You had finished at 7.15am, having worked through the night a 12 hour shift, and you had slept for something like two hours that morning."
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