Pilot schemes will be launched within the next 12 months, probably from three ports of entry - Heathrow, Dover Docks and the Channel Tunnel - to test the system before it is fully introduced.
The scheme will cost pet owners about pounds 200 per animal, but ministers do not believe the fees will lead to increased smuggling of pets.
It will be a big saving on the current average bill of pounds 2,000 for putting pets into quarantine kennels for six months. Pets will be expected to have an electronic identity `chip', vaccination against rabies, a blood test and an anti-rabies certificate.
Pet owners who campaigned for the scheme said it would end the suffering of animals locked up for half a year in quarantine kennels.
"At last, after 100 years of compulsory quarantine for all pets arriving in Britain, pet owners will be able to travel in Europe and rabies-free islands without being separated from their pets for six months as soon as they return," said Lady Fretwell of Passports for Pets.
It was described as a "momentous occasion" by Chris Laurence, the acting chief veterinary officer of the RSPCA. "Thousands of pets will be spared the unnecessary stress of being separated from their owners and placed in solitary confinement," he said.
Announcing the scheme, Nick Brown, the Minister of Agriculture, said he was confident the checks would provide the same level of protection against rabies spreading to Britain as the quarantine laws.
The scheme follows the recommendations of a study by Professor Ian Kennedy of University College London. There will be pre-entry checks at ports and airports, with random spot checks to police the new system.
It will cover the European Union, Monaco, Andorra, the Vatican, Liechtenstein and Switzerland, as well as rabies-free islands such as New Zealand, Japan, Barbados, Jamaica and Australia. But it will exclude Cyprus because the north, held by Turkey, has rabies. Talks are continuing over including Canada and the US.
Airlines and ferry companies will be expected to carry out checks to ensure the certificates match the pets, and qualified vets will check to make sure the vaccinations have worked.
Pet-loving MPs welcomed the scheme. Robert Key, a former Tory transport minister, said he could take his springer spaniel Tigger on holiday to France. The Labour MP Jane Griffiths said it was good news for the fancy rats she breeds.
There are likely to be VIP's pets that will benefit, including the royal corgies and Whisky and Soda, the two dogs owned by Chris Patten, the former governor of Hong Kong.
Allowing tourists to travel with their pets to Britain will also boost the tourist industry, said Denis MacShane, the MP for Rotherham.
The losers will be the owners of quarantine kennels who are facing a sharp downturn in business. Mr Brown said he could not offer them compensation.
But their business will not be completely finished - pets refused entry, or those from countries not covered by the scheme, will still need quarantine kennels.