The Tahiti Nui is expected to travel for four days to reach Nukualofa, bringing water, storage tanks and other material as part of an international relief effort.
The French Polynesian government contributed following Tonga's appeal for assistance, relayed via the Pacific Islands Forum.
The ship is carrying wood and tarpaulins to help build temporary accommodation for families and support reconstruction.
The cargo also includes clothes, sanitary products and non-perishable foodstuffs, collected during a week-long drive supported by the churches
The French Polynesian president Edouard Fritch said the outpouring of solidarity that characterises Pacific peoples has not failed its reputation for generosity in the most difficult situation.
The French military had earlier sent one patrol boat with 40 tons of supplies from French Polynesia and another one with 10 tons from New Caledonia.
It also flew in supplies from Noumea in an operation with the French foreign ministry, liaising with the European Union and the Tongan authorities.
Because of Tonga's Covid-19 restrictions, all deliveries are being made without any contact with the local population.