Sarrazin, who won at the venue last year, was transported to hospital by helicopter after losing control during his second practice run on Friday, bouncing off the surface and sliding a long way.
"Cyprien Sarrazin underwent surgery last night to decompress an intra-cranial haematoma," the FFS said in a statement.
"The operation went well, but he remains under anaesthetic for the time being. We will give you more information later today."
Sarrazin won three downhill races in the previous World Cup, along with a Super-G race in January to finish fifth overall.