The zoo said fire crews were working to control the blaze and staff were working to move animals out of danger.
One eyewitness, David Clough, said high winds fanned the flames in the inflatable roof of the building, adding: “We were very worried for the people and animals that would have been in the building.”
The zoo said in a statement: “Visitors have been evacuated and asked to leave as teams work to bring the situation under control.
“The zoo’s animal teams are working to move all animals away from the incident.”
Mr Clough, 50, who lives across the road from the zoo, said: “We first saw signs of the fire shortly after 11.30.
“The Monsoon Forest building in the zoo’s new Islands development was on fire.
“It has an inflatable roof that was burning. Lots of flames and smoke.”
He added: “Orangutans and gibbons are our nearest neighbours there, but there are many other animals, including free-flying birds.
“It was spreading across the roof in strong winds for a while.
“Lots of fire engines arrived quickly.
“By about 30 mins ago the main flames had gone, but there’s a smaller fire still burning at the southern end of the roof.”
A zoo visitor who declined to be named told the Press Association: “We were in the Monsoon enclosure when it happened and were rushed out due to an electrical fire. It spread very quickly.
“The staff ensured all the members of the public were safe. Many staff (were) running towards the fire, (I’m) assuming to help with evacuating animals.
“The entire zoo is closed - a shame as we’ve travelled up from Plymouth.”