Seven fire engines and an aerial appliance were called to the property, which is above a shop, on Chapel Street in Leigh at 9.20am on Sunday.
No-one was in the flat at the time and the animals could not be saved.
Two people were helped out of a neighbouring property and treated for minor smoke inhalation.
Chapel Street was closed for several hours while firefighters worked to bring the blaze under control.
A significant amount of damage was caused to the flat, but the flames did not spread to any other properties.
A fire investigation was launched to find out what happened and it is believed to have been an electrical fire which started accidentally.
Fire crews remained at the flat until around 3pm.
Chapel Street, which had been closed between Astley Street and Warrington Road, has since reopened to traffic.