Akbari, 61, was hanged after being convicted of "corruption on earth and harming the country's internal and external security by passing on intelligence", the judiciary's Mizan Online website said. It did not say when or where the execution took place.
Mizan Online said Akbari, who had been arrested more than two years ago, had been a spy for Britain's MI6 secret intelligence agency and had received around $2 million for his services.
British Prime Minister Rishi Sunak said he was "appalled" by the execution.
"This was a callous and cowardly act, carried out by a barbaric regime with no respect for the human rights of their own people," Sunak tweeted, adding his thoughts were with "Alireza's friends and family".
UK Foreign Secretary James Cleverly warned on Twitter that his execution "will not stand unchallenged", a statement echoed by France which condemned the execution "in the strongest terms" and said it cannot go "unanswered".
Iran summoned the British ambassador to protest what it described as "unconventional interventions", after the UK said it would summon Tehran's envoy.
Akbari was hanged only hours after the United States had joined its ally Britain in calling for Iran not to go ahead with the execution.
US diplomat Vedant Patel said on Friday that Washington had been "greatly concerned" by reports that Akbari had been "drugged, tortured while in custody, interrogated for thousands of hours and forced to make false confessions".
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