French forces have taken control of the airport in the Malian town of Kidal, the last remaining urban stronghold of Islamists in the north, as France's foreign minister vowed troops would depart from Mali "quickly".

Kidal, 1,500km from the capital Bamako and towards the Algerian border, would be the last of northern Mali's major towns to be retaken by French-led forces after they reached Gao and Timbuktu earlier this week in a campaign to drive out al-Qaida-linked Islamists.

The military operation in Kidal itself was "ongoing" on Wednesday, according to French armed forces spokesman Colonel Thierry Burkhard in Paris. The airport was secured on Tuesday night by French troops.

"They arrived late last night and they deployed in four planes and some helicopters," said Haminy Belco Maiga, president of the regional assembly of Kidal.

"Afterwards they took the airport and then entered the town, and there was no combat," said Maiga, who had been in touch with people in the town by satellite phone as all the normal telephone networks were down. "The French are patrolling the town and two helicopters are patrolling overhead." Kidal is the capital of the desert region of the same name, to which Islamist fighters are believed to have retreated during nearly three weeks of French airstrikes and an advance by hundreds of ground troops.

Tuareg rebels of the Mouvement National de Liberation de l'Azawad (MNLA), who want greater autonomy for the desert north, said this week that they had taken control of Kidal after Islamists abandoned the town.

The MNLA, which fought alongside the Islamists before being sidelined by them in mid-2012, was not immediately available for comment on the French deployment.

Earlier this week, after French and Malian troops secured Timbuktu, French president François Hollande said: "We're winning this battle, and when I say 'we', I mean the Malian army, the Africans backed by the French."

France, which has 3,500 troops on the ground, as well as fighter planes and helicopters in the country, is cautious about declaring victory, knowing that Islamists who have retreated to desert hideouts could stage comebacks that would be difficult to contain for Mali's weak army. But France has insisted on the rapid implementation of a European training programme for the Malian army as well as the deployment of a joint African force which will eventually assume control of operations.

"Now it's up to African countries to take over," France's foreign minister, Laurent Fabius, told Le Parisien on Wednesday. "We decided to put the means in – men and supplies – to make the mission succeed and hit hard. But the French aspect was never expected to be maintained. We will leave quickly."