The blaze broke out in the roof of the Monsoon Forest area, leading to an evacuation of visitors. The zoo said the cause of the fire was unknown.
Orangutans, macaques, gibbons and larger birds were saved, but some frogs, fish, insects and small birds were lost in the fire.
The zoo said it was “one of the toughest days in its history”.
In a statement, it said: “It’s absolutely heart-breaking to lose any animal, especially when conservationists have worked so hard to breed these wonderful species.”
The tourism attraction, which has more than 21,000 animals, said all the creatures led to safety were being relocated within the site.
Jamie Christon, the zoo’s chief operating officer, said: “Yesterday was one of the toughest days in Chester Zoo’s long history.
“Keepers were able to encourage all mammal species away from the fire and to safety – including the zoo’s group of critically endangered Sumatran orangutans, Sulawesi macaques, endangered silvery gibbons and birds such as rhinoceros hornbills.
“We are though, devastated to say that we were unable to save some of our insects, frogs, fish and small birds who were located near to the outbreak of the fire.”
The site reopened to visitors on Sunday, but a zoo spokesperson said the Monsoon Forest area and Islands zone would remain shut.
Fifteen fire crews and ambulance staff attended after the blaze broke out just before 11:30 GMT on Saturday.
One person was treated for the effects of smoke inhalation.
The zoo said an investigation would take place over the coming weeks.
It set up a fundraising website, appealing for £50,000 towards its conservation work – a target achieved in little more than 24 hours since the blaze began.
The Monsoon Forest habitat is the UK’s largest zoological building, according to the attraction. It opened in August 2015.
The 14-acre section has its own climate, with temperatures reaching 26.6C to replicate conditions in South East Asia.
Mr Christon said: “The strength and support from the public has been incredibly overwhelming and the messages of goodwill have been of great comfort to our teams.
“We will support each other in rebuilding this part of the zoo and continue our mission of preventing extinction.”