The adult male Heck Cattle bull was on the loose for about 40 minutes before it was tranquillised by a team of keepers and vets.
Visitor John Monahan said: "We were told to go into the chimpanzee compound. We were there for about an hour.
"People were in a bit of a panic, running about.
"But the staff were really on the ball and helped people off the grounds as soon as possible.
"They made sure everyone was inside. It was all quite scary.
"A huge bull running about the place – we just came to see the pandas."
Rory Davidson, 30, who lives in the capital, was visiting with his girlfriend Melissa Japp when he saw two bulls fighting in their enclosure.
The couple were taken to safety and held with other visitors in the zoo's cafe. The couple only learned later that one of the bulls had escaped.
Mr Davidson said: "I'm from a farm so I'm used to seeing that sort of thing, but they're pretty big beasts nonetheless.
"One of the staff came along and said it was in a bit of distress so we moved away, and it was only later that we heard it had got out.
"They kept us in the cafe until it had been caught. It's not quite what you expect from a day at the zoo."
A zoo spokeswoman said: "The bull stayed within the locality of the enclosure.
"Edinburgh Zoo visitors were immediately escorted to indoor areas in the zoo. Edinburgh Zoo's trained team of expert keepers and veterinarians safely and effectively darted the animal.
"Staff are establishing the circumstances of the escape, and have secured the area. We regret any inconvenience this caused and our visitor services staff will respond to any queries related to this incident."
The incident comes less than a week after a bird escaped from the zoo.
The scarlet ibis was recaptured after evading its keepers for six days.
It left its enclosure through a hole in the netting on August 22 and its bright red plumage was spotted across the capital.
Zookeepers tracked the two-year-old female to the Cramond area last Sunday and tempted the bird into a trap using her favourite food of mussels and prawns.
The latest escaped animal was part of the zoo's herd of Heck Cattle, which sport horns almost three feet long.
Three bulls and nine cows arrived at the zoo from Devon in 2009 and are located in a field near the top of the hill on which the zoo is located.
In May, hundreds of visitors took shelter after a hog family escaped from keepers and ran wild. Guests sought sanctuary in the monkey house as staff with sweeping brushes and dart guns pursued the animals.
The hogs – two adults and two young – were later captured.
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