Dash-cam footage released by Greater Manchester Police shows a pursuit beginning in the Abbey Street area of Leigh, just after 10pm.
A black Peugeot car is then seen to drive off along Leigh Road towards Atherton.
The car is filmed reaching speeds of up to 80mph and veering onto the wrong side of the road.
Police say the car continued to be driven at speed around Atherton before it entered the Hag Fold estate and pulled up in Spa Road.
The car was then put in reverse and struck the officer, causing an injury to his hand.
Moments later the vehicle was abandoned. One 17-year-old was arrested in the immediate vicinity but a second man, inside the car, is thought to have fled the scene.
A Greater Manchester Police spokesman said: "Shortly before 10.10pm last night, a police car was rammed by a black Peugeot car, which then drove off in the direction of Atherton.
"The police engaged in a pursuit of the vehicle and a man was arrested. The officer sustained broken fingers as a result of the incident."
The force helicopter was also deployed as part of the operation and police dog handlers.
Later the affected officer was taken to Wigan Infirmary for treatment to his hand.
People who know anything about the incident are being asked to call the non-emergency number, 101, quoting log number 2195 of February 12, or Crimestoppers, in confidence, on 0800 555 111.