The animal was one of eight big cats kept at the industrial site near the German town of Schkeuditz, according to local media.
In a statement to the BBC, police said officers shot the animal, a male tiger, to eliminate any further risk to the public.
It added that the local police department received the emergency call at around 12:50 local time (11:50 BST) on Sunday, with officers arriving at the scene quickly.
A spokesperson said how the tiger escaped is part of an ongoing investigation.
No other animals have escaped from the site and police have planned a drone search to ensure it is fully secure.
District Mayor Thomas Druskat called for the other animals to be relocated and told local media it was “unthinkable” what could have happened if anyone else had been harmed.
Nearby residents told the Deutsche Presse-Agentur (DPA) that the incident was “terrible and worrying”, with one claiming the animals were not kept in appropriate conditions.
Animal rights charity Peta called on the government to act, telling DPA that stricter rules needed to be put in place to protect privately-owned animals.