He beat a Conservative rival by 170 votes.
The Labour candidate came third with eight fewer votes than the Conservatives.
Labour, which has an overall majority of 21 on the council, said it was "obviously disappointed".
The turnout for the election was 22%.
Alan Olive, regional director of the Eastern Region Labour Party, said: "Nasty, extremist and racist, the BNP got in narrowly on the back of empty promises, low turnout and bogus respectability.
"The truth is they have absolutely nothing to offer and will let local people down."
Mr Geri, 47, whose grandfather was Italian, said he was "pleasantly surprised" by the result.
"We worked hard for it," he said. "We bring common sense. Things have gone the wrong way. The way people don't want to see them go and that is why we voted for us.
"The asylum problem, NHS problems, crime and the causes of crime".