Graham Manson, spokesman for the Real Fathers 4 Justice group said two members of the organization were involved in the demonstration, rushing forward from seats in the studio audience.
Protesters Ray Barry and Nadine Radford carried out the protest to denounce what they said is "the lottery faced by people every day in the family law courts," Manson said.
"They chose the show to protest because it is live, high-profile and watched by so many people. They also wanted to highlight the fact that so many people find themselves on the losing side of the law court lottery," he said.
An investigation has been launched into the incident, a BBC spokeswoman said, speaking on condition of anonymity.
She said the lottery draw was held up for a few minutes, before the demonstrators - who could be seen by viewers - were removed from the studio.
London's Metropolitan police said officers had not been called to deal with the incident.
Real Fathers 4 Justice is a splinter group of a defunct organization that campaigned for changes to child-custody laws.
The original group staged a series of attention-grabbing stunts - including hurling purple powder at Prime Minister Tony Blair in the House of Commons - but said it was disbanding in January following allegations extremist members were plotting to kidnap Blair's five-year-old son Leo.