Juma was a jaguar at CIGS zoo, which is part of the army's jungle warfare-training centre and where many of the animals have been captured by soldiers on patrol.
A guest who attended the event said: "The jaguar was in a secluded spot, but everyone took pictures with her.
"When the event was over, Juma was taken to her cage, which was in a truck. It was then she ran away."
Physiotherapist Igor Simoes, who attended the event, added: "The jaguar played with me and was on my back.
"However, when she returned to the truck at the end of the ceremony, something happened and she got away."
A statement issued by Brazil's Amazon Military Command said: "Efforts were made to capture the animal by firing tranquilizers. But even though the animal was hit, it still advanced towards a soldier that was stationed nearby.
"As a safety procedure and aiming to protect the soldier and the handlers, a pistol was used to shoot the animal. She died at the scene."
An inquiry has now been launched by the Amazon Institute of Environmental who said it had not given permission for Juma to be used at the event.
It had given permission for another jaguar, Simba, to go show at the event. She was held elsewhere on the courtyard by members of the army.
A spokesman for the agency said the Brazilian army had been asked for an official explanation into the death of Juma and the reason she was "exposed to the Olympic torch event without authorisation".
If the army is found to have breached environmental regulations it could be fined between NZ$1,230 and NZ$123,000.
The shooting of the jaguar was immediately met by an outpouring of anger on social media, with people accusing the army of acting "irresponsibly" and "incompetently".
One said: "The cat was a victim of human stupidity."
Another comment read: "You could see from the event that the poor animal was so stressed out it should never have been there...this is the saddest thing".
The jaguar is already an endangered species in the Americas, with a 30 per cent reduction over the same number of years.
During that period, the jaguar population declined by at least 10 per cent in the Amazon.
The shooting comes just weeks after a gorilla, called Harambe, was killed in an American zoo after a boy fell into its enclosure.
The Olympic torch relay will pass through more than 300 towns and cities from the Amazon to Brazil's southern border, arriving at the Maracana Stadium in Rio de Janeiro on August 5.
