The striker became Cardiff's record signing when a deal was struck with the Ligue 1 club in January.
Cardiff claimed the agreement to pay Nantes a £15m ($18m) fee was not legally binding because conditions insisted upon by Nantes were not fulfilled.
The FIFA Players' Status Committee said on Monday the €6m Nantes should receive from Cardiff is "corresponding to the first instalment due".
A statement read: "In a meeting held on 25 September 2019, the FIFA Players’ Status Committee established that Cardiff City FC must pay FC Nantes the sum of EUR 6,000,000, corresponding to the first instalment due in accordance with the transfer agreement concluded between the parties on 19 January 2019 for the transfer of the late Emiliano Sala from FC Nantes to Cardiff City FC.
"The FIFA Players’ Status Committee, which never lost sight of the specific and unique circumstances of this tragic situation during its deliberations on the dispute at stake, refrained from imposing procedural costs on the parties.
"The findings of the decision were notified to the parties concerned today. Within a deadline of ten days, Cardiff City FC and FC Nantes can request a copy of the grounds of the decision, which can be appealed to the Court of Arbitration for Sport (CAS) in Lausanne."
The exact total amount FIFA has ruled must be paid across further instalments has not yet been revealed, while neither side have made public their intention to appeal the Players' Status Committee's decision with CAS at this juncture.
Investigations into the exact cause of the player and pilot's death have been ongoing since the tragedy took place, with a 64-year-old man from the North Yorkshire area having been arrested on suspicion of manslaughter by an unlawful act earlier this year.
In addition, a man and woman were jailed in September for illegally accessing and circulating the late Argentine's post-mortem photos.