The 52-year-old coach had been told by president Jean-Michel Aulas that he would be offered a two-year contract extension if Lyon reached the French Cup final and finished in the Ligue 1 top three.
But a 3-2 Cup semi-final defeat by Rennes on Tuesday put his future in doubt, and fans on Saturday unfurled banners calling for Genesio’s departure.
“We had already been eliminated on Tuesday. It’s unacceptable,” said goalkeeper Anthony Lopes, who added that the players need to put Genesio’s unstable situation to the back of their minds.
“We are professionals. We must know what to do. We know that it’s football. Part of the decision on the (future of the) coach was made on Tuesday but we must know how to ignore all that.
“The truth is on the pitch. If we don’t turn up on the pitch it can only be difficult in the future.”
Martin Terrier’s first-minute goal briefly placated the home support, but only six minutes later Lyon were behind.
Wesley Said danced through the hapless home defence to slot in a third-minute equaliser, before his speculative long-range effort was deflected past Lopes by Lyon centre-back Marcelo.
Genesio’s beleaguered men never seriously threatened a response and put paid to their own hopes in comical circumstances midway through the second half.
Said’s pull-back was inadvertently backheeled into the net by Lyon full-back Rafael, with the spin on the ball leaving Lopes grasping at thin air.
A first league win since January lifted Dijon off the foot of the table and into the relegation play-off place.