Stade Reims were another top-flight team dumped out when they lost 1-0 at amateurs Plabennec who also upset Ligue 1 sides Nice and Nancy on their way to the last 16 in 2010.
Ajaccio, fifth from bottom in the league, were shorn of five players on international duty ahead of this month's African Nations Cup while striker Adrian Mutu was dropped.
Former Marseille coach Emon, who replaced the sacked Alex Dupont on December 23, watched his new side struggle throughout and they needed an 89th-minute equaliser from Wilfried Zahibo to make it 1-1 and set up extra-time.
There were no more goals in the additional half hour and Ajaccio were then thwarted by Rouen goalkeeper Theo Defourny in the shootout as he saved three penalties.
"We have achieved something beautiful. I just tried to relax and luck was on my side," Defourny told French radio RTL.
"I think we deserved to win. We had more goal chances."
While most of the top-flight clubs involved on Saturday eased past lower division sides, Steven Coat's first-half goal was enough to give fourth tier Plabennec a 1-0 victory over Stade Reims.
Nancy, bottom of Ligue 1, crushed sixth tier Dreux 5-1 while fifth-placed Lorient thrashed amateurs Chauray by the same score.
Ligue 1 leaders Paris St Germain will visit fifth tier Arras on Sunday with a revamped squad after coach Carlo Ancelotti decided to rest Zlatan Ibrahimovic, Thiago Silva, Thiago Motta and goalkeeper Salvatore Sirigu.
Also on Sunday, Lyon travel to third tier Epinal and Marseille host 2009 winners Guingamp