Robert Semple was being held by al-Qaeda, the United Arab Emirates state news agency said.
He was freed by UAE forces and is "safe and well", Foreign Secretary Philip Hammond said.
The oil industry worker, in his 60s, is receiving support from British authorities, Mr Hammond added.
Abu Dhabi Crown Prince Mohammed Bin Zayed Al Nahyan informed David Cameron on Saturday, the WAM agency said.
The agency reported Mr Semple had spoken to his wife by telephone and would return to the UK after medical checks.
The Foreign Office has not officially confirmed his name.
The UK's prime minister tweeted: "I'm so pleased for the family of the British hostage in Yemen - who has been released safe and well. Thanks to the UAE for their help."
Mr Hammond added: "We are very grateful for the assistance of the UAE."
Yemen has seen conflicts between several different groups in recent months, pushing the country "to the edge of civil war", according to the UN's special adviser.
The main fight is between forces loyal to the President, Abdrabbuh Mansour Hadi, and those allied to Zaidi Shia rebels known as Houthis, who forced Mr Hadi to flee the capital Sanaa in February.
Al-Qaeda in the Arabian Peninsula (AQAP) has also staged numerous deadly attacks from its strongholds in the south and south-east of the country.
Iona Craig, a freelance journalist in the country, told BBC News she was in Sanaa when Mr Semple was taken hostage last February.
She said he had been forced from a car by gunmen and had been missing since.
At the time, there were attempted kidnappings on "an almost weekly basis" in the city, she said.
There are "very few" foreign workers left in the country, Ms Craig added.
She said UAE troops were based in the south, suggesting Mr Semple was being held or was freed in south of the country.
Several Britons have been kidnapped in the country.
UK-born US journalist Luke Somers, 33, was killed by al-Qaeda militants last December as US special forces attempting to free him and South African teacher Pierre Korkie.
He was kidnapped in September 2013 in the capital Sanaa while working as a photojournalist.
In July 2014 Mike Harvey was released five months after being captured, also in Sanaa.
A British national being held hostage in Yemen has been released following a military operation, the Foreign Office has announced.