South Yorkshire Police said PC Dave Fields, 45, was responding to an incident on Monday when the marked police BMW he was driving was in a collision with a Citroen in which Lorraine Stephenson was a passenger.
PC Fields' family said in a statement: "Dave was a loving husband and dad of two, who was a dedicated officer committed to his job.
"We are heartbroken by our loss."
PC Fields and Mrs Stephenson suffered fatal injuries and were pronounced dead shortly after the crash at 8.15pm on Christmas Day.
It happened as the police car was travelling on the A57 towards Coisley Hill.
Police said the driver of the Citroen, a 63-year-old man, was taken to hospital where he remains in a serious condition.
The incident has been referred to the Independent Police Complaints Commission.
Police said specialist officers were working to support both families.
On Tuesday, Assistant Chief Constable David Hartley said: "We have lost a friend and a colleague from our police family in this incident.
"The officer has been with us for 12 years and was a passionate, professional and universally liked officer.
"His colleagues and everyone across the force are devastated by what has happened."
(c) Sky News 2017: Tribute to 'dedicated' officer killed in Christmas Day road crash
