An escaped hippo was cornered in one of the city’s main squares and subdued with a tranquilizer gun, the zoo said.
Some other animals also have been seized, but it remained unclear how many are on the loose. Bears and wolves are also among the animals who fled enclosures amid the flooding from heavy rains and high winds.
There were no immediate reports that any of the fatalities were due to animal attacks. The zoo said one of the dead was Guliko Chitadze, a female zookeeper who lost an arm in an attack by a tiger last month. Her husband also died in the flooding.
Heavy rains and wind hit Tbilisi during the night, turning a normally small stream that runs through the hilly city into a surging river. The flooding also damaged dozens of houses.
City mayor David Narmania told journalists that 10 people were known to have died; 10 others are reported missing.
Mayor Narmania said: “Not all the animals who ran away from the zoo have been captured. Therefore, I want to ask the populace to refrain from moving about the city.”
Helicopters are circling the city and residents have been told to stay indoors, except in an emergency. About 1.1 million people live in the former Soviet republic’s capital.
By mid-afternoon, it was still unclear how many animals remained on the loose or what species they are.
A full accounting of what animals were missing was not possible because a large part of the zoo remains underwater, zoo spokeswoman Khaati Batsilaishvili said.