The spokesman further said that the agent had confessed to the his/her cooperation with the British-based cultural council and had transferred good intelligence to the British security services and had been sentenced to 10 years in prison.
Esmaili did not identify the suspect by name or gender.
Esmaili added that the suspect, whom he alleged had been tasked with drawing up and managing cultural "infiltration" projects, had been arrested by Iranian intelligence and security agencies "more than a year ago".
A friend of the sentenced woman identified her as Aras Amiri to the Reuters news agency, while British newspaper The Telegraph reported that one of Amiri's family members confirmed she was the person sentenced.
It was not immediately clear if she was the person whose sentencing was announced on Monday.
Iranian authorities accused her of plotting against the government and handed her a five-year jail sentence. Funding from the UK's foreign office makes up about 15 percent of its overall income.
Pilot pulls off nail-biting landing despite front wheel failure
The incident came less than a week after a Biman Bangladesh Airlines skidded off the runway while landing at Yangon airport. He says the pilot informed the airport before landing that he was unable to pull down the nose wheels.
Nazanin Zaghari-Ratcliffe with her daughter, Gabriella, in 2016.
Ciaran Devane, the British Council's chief executive, said the organisation had not yet "been able to confirm" whether the unnamed woman cited by Esmaili was an employee.
Mr Esmaili said such actions "drew the attention of Iran's security and intelligence services", which led to the woman's arrest a year ago.
The spokesman said the convict has confessions about the process of being recruited by and cooperating with the so-called cultural council of the United Kingdom, trainings he received from the United Kingdom secret service, and his missions to carry out cultural projects in the country.
The British Council said past year that it was aware that one of its staff had been detained while making a private family visit, without identifying them as Ms Amiri.
"Our colleague's safety and wellbeing remain our first concern, as it has been throughout their detention", he said, according to the Independent. "British Embassy officials in Tehran are in touch with the Iranian government to seek further information".
"We are a non-political organisation committed to people-to-people engagement, and our staff are not connected to any espionage agency".