Air Carrier Captain reported after parking at the gate and upon exiting the aircraft; outgoing flight crew reported engine fire on ramp to pilots. The Captain returned to the flight deck; pulled engine fire handles; obtained a fire extinguisher; and went to ramp. The engine tailpipe fire had subsided; and remaining fire was extinguished by the Captain.

Narrative

Normal; approaching; landing; taxi to the gate; and engine shutdown. The First Officer and I were gathering our bags to leave the aircraft. The outgoing Captain had just come up from the walkaround and indicated to me that we had the Number One Engine on fire. I told the A Flight Attendant to expedite the deplaning. I went back in the cockpit and told the First Officer to call for the fire trucks. I pulled the Engine Fire switches and verified that the engines were shut down. I then grabbed the cockpit fire extinguisher and went down to the ramp.According to the Captain; the fire had subsided quite a bit. I waited about 45 seconds; and then I extinguish the fire with the handheld fire extinguisher. The fire went out. Manager; and other Supervisors arrived along with the Fire Department and Maintenance. I went back in to the jetway and ensured that the passengers were all deplaned. I called Dispatch; and made a logbook entry for both the engine fire and the use of the fire extinguisher.