Prince Philip, 97, was not injured in the accident, which happened as he drove a Land Rover out of a driveway onto the A149 shortly before 15:00 GMT.
The other car involved was a Kia. Two women in it needed hospital treatment - they have since been discharged.
Eyewitnesses said the duke's Land Rover overturned during the collision.
They said they helped the duke out of the vehicle. He was conscious but very, very shocked and shaken, they added.
Norfolk police said it was force policy to breath test drivers involved in collisions and both had provided negative readings.
The duke is back at Sandringham and has seen a doctor as a precaution.