The female jaguar escaped its handlers after the ceremony on Monday and attacked a soldier, a spokesman said.
Four tranquiliser darts failed to stop it and a soldier shot it with a pistol.
Organisers for the Rio Games said it had been a mistake to exhibit the Olympic torch next to a chained wild animal.
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University of Brasilia animal behaviour scientist Joao Paulo Castro told BBC Brasil that it was likely the animal escaped because it got stressed during the relay.
"It's neither healthy nor advisable to subject an animal to such a situation, with lots of noise and people," he said.
"Often, jaguars already are stressed by being kept in captivity, that's only compounded when they're exposed to hubbub."
Animal rights groups have condemned the killing, with some questioning why the animal was involved in the Olympic event.
"When will we learn? Wild animals held captive and forced to do things that are frightening, sometimes painful, and always unnatural are ticking time bombs," Brittany Peet, director of captive animal law enforcement at People for the Ethical Treatment of Animals (PETA), said in a statement.
The animal called Juma was being kept at a zoo attached to a military jungle training camp.
It was rescued as a cub by soldiers after its mother died in the jungle and was brought up at the camp, the military said.
Jaguars seized from hunters are sometimes kept as mascots by jungle battalions and shown at military parades.
An investigation into the jaguar's death will now be carried out by the military.
The Olympic torch is relayed through Brazil leading up to the August opening ceremony.
A jaguar has been shot dead shortly after it was used in an Olympic torch relay in the Brazilian city of Manaus, the army said.