The animals managed to climb across a fallen tree that had come down in strong winds and attacked a wild boar in the neighbouring section.
Bosses at Whipsnade Zoo in Dunstable, Bedfordshire, said they had no choice but to euthanise the bears over concerns of an “immediate threat to human life”.
The zoo’s chief curator, Malcolm Fitzpatrick said: “As brown bears are strong and dangerous predators, our first priority is safety.
“We must quickly make decisions informed by our experience and expertise to protect our people, guests and our other animals.”
He said tranquilisers were not an option as they would have taken at least 20 minutes to work, “during which time the animals can become unpredictable and aggressive as adrenaline is coursing through them”.
“We had to intervene immediately to ensure the bears didn’t get out of the boar enclosure, which has a low fence,” he added.
A third brown bear had not left the enclosure, and was called to her indoor den and secured while the tree was removed.
Mr Fitzpatrick said there would be a full investigation into Friday’s incident, adding: “As zookeepers and animal carers, this situation is something we train to deal with through regular, rigorous drills – but one that we always hope we’ll never have to face.
“I’m devastated by the outcome of this morning’s incident, but I’m confident that our actions prevented any further loss of life.”