It was not long before Rimington realised that the Russian man was from the KGB, the Soviet security service.
She said: "By that time I was the clerk typist in the MI5 office and alert to this sort of thing and it seemed to me to be an obvious effort to introduce us to somebody from the Soviet Union.
"I realised, quite quickly, this was a man from the KGB because, being the clerk typist in the MI5 office, we had a list of those people we had identified as KGB officers - and he was one of them.
"It was quite clear this was the beginning of a chatting-up operation and it might lead anywhere. We didn't meet those people again."
She added: "They didn't know who we were, but I knew who he was."
Rimington would go on to become the first female director general of the Security Service, having been one of the earliest women to break through into front-line intelligence work. Judi Dench's character M, who was the head of MI6 in the James Bond films, is allegedly modelled on Rimington.
She had been given the "tap on the shoulder" in 1967 by one of the first secretaries to the commission in Delhi, who was MI5's man in India.
For the next two years she served him, before returning to London in 1969 and taking on a full-time job at MI5.
At the time women were "second class citizens" in the service, she said.
Rimington and female colleagues started a "quiet revolution" to bring women into intelligence roles and she quickly rose through the ranks.
In 1992, when given the top job, she became the first director general to be named publicly.
Rimington said: "Strangely enough, the time I feared most for my personal safety and that of my family was when my name was announced. Someone had taken a photograph of our house and put it on the front page of one of the newspapers.
"We had to move - quite frankly, we had to leave our house overnight."
Rimington, who now writes novels, was speaking after an event to launch a National Archives exhibition about the Cold War, called Protect and Survive.