The New Caledonian Army Forces were quick to act, sending a patrol plane last Friday to look at the extent of the damage in Vanuatu.
On Sunday the territory's forces arrived on the cargo ship D'Entrecasteaux in the capital Port Vila.
A supply ship is also delivering supplies from Port Vila to Tanna and other islands.
The ship will assist in road clearance, provide drinking water and give medical response.
General of the Army Forces of New Caledonia Valerie Putz says the structural damage on Tanna Island has been fairly minimal.
"I know that Vanuatu authorities have expressed their concerns with power outages but it seems to have been put back in place. Water supply was a concern as well.
"There is one French army detachment of engineers who are in Tanna to restore the viability of the roads.
"There is also one helicopter that was deployed last night (Tuesday), ready to offer transport help to Vanuatuan authorities or to have a damage assessment of other islands in Vanuatu."
On board the ship, launched from New Caledonia, were four tonnes of humanitarian material, two four-wheel drive military vehicles, a rescue team from the territory's civil security, and military personnel.
RNZ Pacific reports a fresh water commercial company has donated eight fresh water tanks as part of the cargo.
The operation was launched by the New Caledonian High Commission.
Putz said there has been great cooperation between both New Zealand and Australian defence forces.
"There is also a deep co-ordination between the New Zealand Defence Force and the Australian defence force which has been led under France's diplomatic mechanism of co-ordination.
"At a ground level, the Australian and New Zealand forces use the French ship to bring humanitarian resources to Tanna Island."
Meanwhile, the HMAS Canberra was due to arrive by Thursday morning with more than 600 Australian Defence Force personnel on board along with humanitarian supplies.
According to the ABC, a small Australian rapid assistance team is already in the country and Australian aircraft are conducting aerial surveillance of the worst-affected areas.
Coordinator Barnaby Caddy said the rapid assessment team comprised 12 members skilled in humanitarian and damage assessment, hazardous materials, power, health and infrastructure.
Photo credit: French government Caption: French aid arrives in Tanna