Goalscoring chances were few and far between during the opening 45 minutes, with both defences on top.
Lyon's lead lasted just four minutes, though, as Chahir Belghazouani found the net from the left side of the penalty area to level proceedings.
Ajaccio began to take control of the contest and Mutu's first intervention arrived with 65 minutes played. Johan Cavalli sent over a cross for the Romanian, who headed in from close range.
He went on to make sure of the outcome in stoppage time when he scored from the penalty spot after Lyon's Dejan Lovren had been dismissed for a foul.