"Per our safety protocol, our staff jumped into action and escorted all guests into indoor buildings," community relations manager Erica Hansen said.
Emergency teams found the Amur leopard, named Zeya, and tranquilized her, Hansen said. Zeya was taken to a holding area.
Hansen said it was not immediately clear how Zeya managed to dart out. No injuries were reported.
Zeya's escape came as the Cincinnati Zoo reopened its gorilla exhibit
10 days after a 3-year-old boy entered the enclosure.
The boy had a 10-minute encounter with Harambe, a 450-pound gorilla, on May 28. A witness told CNN the boy's mother was temporarily distracted by other children when the boy fell into the exhibit.
Harambe pulled the boy across a moat before zoo personnel fatally shot the animal.
Hansen, the Salt Lake City zoo official, said Tuesday's leopard escape is much different than the Cincinnati gorilla incident.
She said leopards are smaller than gorillas, and emergency crews were able to sedate the leopard.
Animal experts have said tranquilizing Harambe would have been a bad idea because it could have posed even more danger to the boy.
"I have been in situations where we have tranquilized gorillas," Zoo Miami spokesman Ron Magill said.
"As soon as that tranquilizer dart hits the gorilla, it's like a tranquilizer dart hit you -- you'd go, 'Ow!' But with gorillas ... they have what's called displaced aggression -- where all of a sudden, whatever's closest to them, they think is the reason they felt that dart. That child could have been in even more severe danger had they tried the tranquilizer dart."