But she told AMs there would be further discussions with police about the methods to be used.
The announcement is in contrast to the policy in England, where badgers will be vaccinated against bovine TB in six pilot areas next year.
Environment Secretary Hilary Benn, making the announcement for England last week, said: "Developing an effective vaccine for bovine TB is only half the challenge. The other is to deploy it effectively."
No date for the start of the proposed cull in Pembrokeshire was announced, but Ms Jones said moves to put the legislation into place to allow it to happen would begin next month.
The minister authorised the scheme last April and said a pilot scheme would take place in a TB hotspot to learn lessons before being rolled out across Wales.
Up to 1,000 badgers could be trapped and shot in north Pembrokeshire under the cull proposal
Outlining plans for the cull on Tuesday, she said there had been "attempts over many years to control this disease and they have failed".
"Each member state is, however, obliged under an EU directive to develop an eradication programme in order to 'accelerate, intensify or carry through' the eradication of the disease," Ms Jones said.
"All of this underlines the necessity of our commitment to pursue TB eradication urgently," she added.
Last year, more than 12,000 cattle were slaughtered due to TB - 52% more than in the previous year.
Wales' chief vet Christianne Glossop said the situation was "unacceptable and unsustainable".
"If you talk to farmers, you see a real level of frustration and desperation in their minds at the moment because of the problem," she said
Dr Glossop said west Wales had the highest incidence of bovine TB in Britain and 60% of cattle being slaughtered because of the disease were in north Pembrokeshire.
She said the cull, which will take place over five years, was just one of a number of measures that would be applied in the targeted area of north Pembrokeshire.
Conservative Preseli Pembrokeshire AM Paul Davies said: "I very much welcome the proposals in the statement and I very much hope these plans will see the end of TB in my area - a disease that has wreaked havoc in the countryside of Wales."
Farming unions welcomed the announcement but Labour Wrexham AM Lesley Griffiths appealed to the minister to abandon the plans.
Ms Griffiths argued the science being used to justify the cull was "at best selective and at worst flawed".
Badger Trust Cymru spokesman Steve Clark said the minister had "ignored the weight of scientific opinion and caved in to bullying farming unions and cull-mad vets".
"By combining badger culling with other TB control measures, Elin Jones will have no idea which particular measure reduces the disease," he said.
"But any resulting fall in bovine TB will inevitably be attributed to badger culling, spelling doom for badgers across both Wales and the UK as pressure mounts to repeat this brutal extermination elsewhere," Mr Clark added.
Campaign group Viva, which has opposed the cull, said it was "a political decision designed to appease farmers and protect the seats of rural politicians".
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