The fire broke out shortly after 6am (local time) at the zoo's Animal Adventure section before spreading to the shop and cafe, near to an area where visitors can handle and feed animals, the zoo and London Fire Brigade said.
More than 70 firefighters took three hours to bring the fire under control while desperate keepers who live on the site in Regent's Park in central London moved animals to safety. Some of the staff needed treatment for smoke inhalation and shock.
Ten-year-old Misha the aardvark, one of the zoo's best-loved animals, perished in the fire and four of the meerkats were unaccounted for and are also presumed to have died.
Taking their name from the Afrikaans word for "earth-pig", aardvarks are large burrowing mammals found across much of Africa.
"We're absolutely devastated that Misha the aardvark has been killed and we're still trying to find out what happened to the meerkats, but at the moment the site where this fire took place is closed down," Dominic Jermey, the zoo's Director General, told BBC TV.
"At the moment we're not certain what has happened to the meerkats but I'm not optimistic at this stage, unfortunately."
Other animals nearby are being monitored by vets, but the zoo said indications were that they were unaffected. It said it hoped to reopen on Sunday.
It is the world's oldest scientific zoo, dating from 1826, and houses 20,166 animals, according to its inventory for 2017.
The fire brigade said it had sent 10 fire engines and 72 firefighters to the scene and added that the cause of the blaze was under investigation.
"The fire mainly involved the cafe and shop but part of a nearby animal petting area was also affected," said Station Manager David George.
"When they arrived our crews were faced with a very well developed fire. They worked incredibly hard in arduous conditions to bring it under control as quickly as possible and to stop it from spreading to neighbouring animal enclosures."
