Jacqueline Wileman, 58, was hit by the lorry on Friday at around 1.40pm on Common Road, Brierley.
It then travelled on to Park View, where it crashed into the front of a house.
Mrs Wileman was pronounced dead at the scene.
Her family, who have released pictures, are being supported by specialist officers.
They said: “We are grateful to everyone for their kind messages of condolence and support so far and would like to ask that you continue to respect our privacy as we come to terms with what has happened and grieve for Jacqueline.
“We would also like to express our thanks to those on Common Road and Park View who assisted at the scene and ask anyone who was there, or who has any information, to please get in touch with the police so they can bring those responsible to justice.”
Police said four arrests were made after Mrs Wileman was hit by the vehicle, previously reported as stolen, before it hit two parked cars and ploughed in to the house.
The HGV was being followed by a police vehicle just before the crash, South Yorkshire Police confirmed.
A referral has been made to the Independent Office for Police Conduct (IOPC).
A police spokesperson said: “At around 1.40pm, a HGV, previously reported as stolen, was travelling along Common Lane, towards Grimethorpe, when an officer in a marked patrol car, travelling in the opposite direction, turned round to follow the vehicle.
“The HGV then travelled on to Park View, where it was in collision with a pedestrian.
“The vehicle then hit two parked cars before colliding with a house on the road and coming to a stop.
Yorkshire Ambulance Service said three people were taken to the major trauma unit at Sheffield’s Northern General Hospital following the crash.
No details of their injuries were released.
The man driving the vehicle was also taken to hospital with minor injuries, police said.
Anyone with any information is asked to call 101 quoting incident number 266 of 14 September 2018.