The near-miss happened at about noon on Monday, as the two marked police vehicles were travelling on the A165 at Lebberston, from Scarborough to an incident in Filey.
A blue Suzuki Swift, driving in the opposite direction, then nearly collided head on with both vehicles, a North Yorkshire Police spokeswoman said.
No injuries were reported and one of the police cars continued to the scene they were called to.
A 19-year-old woman has been interviewed in connection with the near-miss and is currently assisting officers with their enquiries.
But the force has appealed for anyone with information to contact police on 101, quoting 12180206897.