Griezmann drilled in a low shot in the eighth and final minute of stoppage time at Education City Stadium. Referee Matthew Conger ruled out the equalizer following a video review.
The FFF's statement overnight Thursday said the goal was "unfairly refused" but did not elaborate, amid reports it focused on the fact Conger restarted the match then blew the final whistle - after which he consulted VAR and ruled out Griezmann's goal.
Griezmann was in an offside position when a cross came over but then retreated and appeared to be played back into an onside position following a Tunisia defender's deflection when he shot into the bottom left corner.
The FFF said that it had 24 hours from the end of the match to file an official complaint with governing body FIFA.
No further details were given.
It was France's first World Cup defeat since 2014, when it lost 1-0 to Germany in the quarterfinals.
Defending champion France finished atop Group D on goal difference from Australia and will face Poland in the last 16 on Sunday.
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