Bourgoin's status as a Top 14 side was weakened further on Friday after a 13-9 home defeat to play-off chasers Toulon.
Despite a strong first-half showing, which had Bourgoin leading 9-7 at the break, the visitors had too much strength for the home side, who must now hope other results on Saturday go their way.
Benjamin Boyet fired Bourgoin into a 6-0 lead early on, with the home side also helped by a yellow card for Mafi Kefu for a late tackle.
But Felipe Contepomi brought Toulon - for whom Tana Umaga made an appearance - into the lead with a try under the posts, which he converted.
Boyet put Bourgoin back into the lead at the break, but Toulon's pack took control in the second half and ground out two more penalties for Contepomi which were enough to seal the win.