Lassana Diakhaby and Ange Gnaleko missed the crucial spot kicks for the minnows as the 10-time former winners avoided a second straight cup exit to a side from National 2.
Abdoulaye Diaby scored for the hosts after 20 seconds following a goalkeeping howler by Yohann Pele and France midfielder Dimitri Payet equalised after 20 minutes for the visitors who had Japan right-back sent off in the closing stages of normal time.
Pele made up for his early mistake by saving Diakhaby and Gnaleko's efforts before Lucas Perrin slotted the winning penalty.
Marseille were knocked out of the competition by Anrezieux at the same stage last season.