A teenager and four children were among the group that ran into trouble at sea and contacted emergency services for help, they said in a statement.
The migrants, who said they were from Iran, Iraq and Afghanistan, were picked up in the early morning and brought to mainland France, several with hypothermia.
The boats ran into difficulty in choppy waters, and one sank shortly after its occupants were rescued, said the statement.
People are taking ever greater risks to reach Britain from France, which has taken an uncompromising approach towards economic migrants who come to Europe in search of a better life.
Since January, more than 1,400 migrants have been rescued either by British or French coastguards -- more than double the number who tried to cross the busy shipping lane in 2018, according to official French figures.
France has attributed the increase to good summer weather emboldening more to attempt the journey.
Such crossings are dangerous given the heavy maritime traffic in the Channel as well as strong currents and cold water.
The body of an Iraqi migrant who tried to swim across the Channel to Britain was found off Belgium's coast in August.