Four helicopters were being used to drop water on the threatened areas as winds spread the blaze across dry bushes.
Authorities closed roads and students from Cape Town University were forced to evacuate.
One firefighter sustained burns and was hospitalised for treatment but no fatalities have been reported, officials said.
The blaze has also destroyed part of a cafe at the Rhodes Memorial overlooking the city's port.
The fire started early on Sunday near a memorial to politician Cecil Rhodes, located on Devil's Peak, before spreading rapidly up the slopes.