Alexandre Lacazette was sent off late on for Lyon to complete a miserable evening.
Lyon came close to an opener in the 16th minute when Bafetimbi Gomis thought he had given the visitors the lead, but Guillermo Ochoa pulled off a one-handed save at full stretch to the deny the striker.
It was the hosts who went ahead two minutes later, when a defensive-splitting pass saw Arrache run onto the ball in between the two Lyon centre-backs before striking early past an onrushing Anthony Lopes into the bottom far corner.
Ochoa was called into action once again just after half time, when he kept out Clement Grenier's free kick.
However, Lyon found an equaliser in the 56th minute through a low strike into the bottom corner from Malbranque after slowing the pace down just inside the box.
Despite being pegged back, Ajaccio managed to find a second goal 20 minutes from time when Camara was played in on the left side of the penalty area, before opening up his body and curling an effort into the far corner of the net.
The visitors' poor evening was complete in stoppage time when Lacazette was sent off after picking up a second yellow card.
Ajaccio move up to 16th in the table while Lyon remain eighth.