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.
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, had been raised in the zoo in the Amazon since it was a cub along with half-a-dozen siblings.
The Olympic torch is relayed through Brazil leading up to the August opening ceremony.