Keepers at the Whipsnade Zoo in Dunstable euthanised the animals on Friday after they got into a neighbouring enclosure.
Malcolm Fitzpatrick, the zoo's chief curator, said the two female bears - named Snow White and Sleeping Beauty - had attacked a male boar after escaping their own enclosure.
He said it had been an "incredibly upsetting and difficult day" having had to take the decision to kill the bears.
"As brown bears are strong and dangerous predators, our first priority is safety," Mr Fitzpatrick said.
"We must quickly make decisions informed by our experience and expertise to protect our people, guests and our other animals."
He said he and his colleagues "agreed the only option was to euthanise the bears," adding: "When there's an immediate threat to human life, the decision is made for you."
Mr Fitzatrick said tranquilisers could not be used, as they would have taken 20 minutes to take effect, "during which time the animals can become unpredictable and aggressive as adrenaline is coursing through them".
The boar which was attacked was treated by a vet and is currently under observation, Mr Fitzpatrick added in a statement released by the zoo on Friday.
A third brown bear called Cinderella, which had not left her enclosure when the other two escaped, was called to an indoor den and detained safely while staff removed the branch.
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