The group emailed a statement saying it “announces to the world, especially the French government, that it was responsible for the abduction”.
They named the engineer as 63-year-old Francis Colump of French firm Vergnet.
He was snatched from a heavily guarded compound in Katsina state by about 30 gunmen last Wednesday.
A security guard and a neighbour were killed in the raid.
Vergnet, which has a wind power project in Katsina, has confirmed the kidnapping.
French President Francois Hollande said on Friday that a group allied to al-Qaeda was probably responsible, but did not name any specific organisation.
“The reason for [Francis Colump’s] kidnap is the stance of the French government and the French people on Islam and Muslims,” Ansaru said in its statement, AFP and Le Monde newspaper quoted the statement as saying.
It mentioned “France’s major role in the (planned) attack on the Islamic state in northern Mali”, referring to France’s backing for an African-led military force to try to oust Islamists from Mali.
The statement also referred to France’s law “outlawing the use of Islamic veil by Muslim women, which is an infringement on religious rights”.
“We inform the French government that this group will continue launching attacks on the French government and French citizens… as long as it does not change its stance on these two issues,” the Ansaru statement went on to say.
Last month, the British government officially declared Ansaru to be a “Nigeria-based terrorist organisation” aligned with Al-Qaeda in the Islamic Maghreb (AQIM).
It is also believed to have loose ties with Boko Haram, which has carried out a series of deadly attacks in northern Nigeria in its bid to impose Islamic law in the country.
President Hollande told Europe 1 radio in France that the kidnappers were “probably linked to AQIM or the groups which are today in Mali”.
Islamist groups which are linked to al-Qaeda seized control of the whole of northern Mali, including the historic city of Timbuktu, in April.
“We have to be firm when it comes to terrorism, while at the same time maintaining contacts to free” French nationals, Mr Hollande said.
Groups linked to AQIM are also believed to be holding several other French nationals kidnapped in Niger and Mali.