The incident took place on the ground floor of the three-storey building in Broomhouse Wynd yesterday afternoon.
The crew used a hose reel and breathing apparatus to put out the flames. No-one was injured in the incident.
Meanwhile, crews had to free an eight-year-old boy whose finger was stuck in a bike chain. Firefighters were called to Woodside Terrace in Joppa at 4.20pm yesterday to release the boy using bolt cutters.