The amount, equivalent to $11.2 million, was unveiled at the annual conference of the European Bank for Reconstruction and Development (EBRD) in London.
Ukraine has accused Russia of using an explosive drone to damage the containment vault at the power plant in February, in an attack that did not result in any radiation being released.
In 1986, a reactor at Chernobyl exploded during a botched safety test, resulting in the world's worst nuclear accident, which sent clouds of radiation across much of Europe and forced tens of thousands of people to evacuate.
Following the incident this year, France has offered funds "to support the restoration of the key functions of" the cover, referred to as the New Safe Confinement.
The "strike rendered systems designed to ensure the NSC's 100-year lifespan non-operational and significantly increased the risk of further deterioration in the absence of swift emergency repairs", said the EBRD, established in 1991 to help former Soviet bloc nations embrace free-market economies.
Thanks to funds from France, the EBRD said it "will support measures to repair the external cladding to stop the ingress of water.
"In the longer term, work will be carried out to restore the core functionality of the facility to minimise the release of further contaminated dust into the environment," the bank added.
France's pledge adds to the EBRD-managed International Chernobyl Cooperation Account, previously holding 25 million euros in funds.
French President Emmanuel Macron's special envoy for the reconstruction of Ukraine, Pierre Heilbronn, said the contribution "illustrates France's longstanding support for Ukraine.
"French companies played a crucial role in building the NSC (up to completion in 2019), and we hope that this first contribution, which comes only three months after the drone attack, should unlock further pledges from partner countries."
British newspaper The Guardian said the repairs could total tens of millions of euros.
The EBRD on Wednesday said it was too early to precise the cost of fixing the cover.
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