Marius, an 18-month-old giraffe, was put down with a bolt gun early on Sunday, zoo spokesman Tobias Stenbaek Bro confirmed.
The body of the giraffe, which was killed in a bid to prevent in-breeding, will be chopped up and fed to carnivores at the zoo.
"It was put down at 9.20am. It went as planned. It's always the people's right to protest. But of course we have been surprised," Stenbaek Bro told AFP.
The zoo explained that it had no choice other than to prevent the animal attaining adulthood.
Under European Association of Zoos and Aquaria (EAZA) rules, in-breeding between giraffes is to be avoided.
Although Marius is healthy, his genes are already well represented at the zoo and he cannot be taken in by the 300 other EAZA-affiliated zoos.
Castration is considered cruel with "undesirable effects", while releasing him in the wild is thought unlikely to succeed.
The giraffe's impending death sparked outrage online, with more than 5,000 people signing a "Save Marius" Facebook petition. More than 3,000 people signed a similar Danish-language online petition and nearly 24,000 an English-language version.
There were reportedly several attempts made to save Marius.
A Swedish zoo, which is not part of the EAZA network, tried in vain to get Marius transferred, the Expressen daily reported.
And another daily, Denmark's Ekstrabladet, quoted a Danish promoter living in Los Angeles, Claus Hjelmbak, as saying he had found a buyer for the animal.
"One of my close friends, a billionaire, said that he wanted to transfer a few million so we could save the giraffe," Hjelmbak was quoted as saying.
"He could easily have lived in his garden in Beverly Hills, but the zoo director was not interested in a sale. I'm angry," the promoter added.
The zoo had made clear from the beginning of the protest that its policy was not to sell the animals.