Marseille, who have won the Cup a record 10 times but have not triumphed since 1989, struggled on a frozen pitch away to opponents playing in the fourth division.
The last 64 game was a 1-1 draw after normal and extra time and Marseille won the shoot-out 5-4, midfielder Benoit Cheyrou scoring the decisive penalty after an effort by Besancon striker Yohann Rangdet was blocked by Marseille keeper Steve Mandanda.
"My players did what they could in very difficult conditions," Marseille coach Eric Gerets told reporters after a game played in freezing temperatures. "That pitch was really, really hard."
Following Sunday's draw for the round of 32, Lyon will entertain Marseille, providing the seven-times champions win their round of 64 tie against Concarneau, postponed twice this weekend because of a frozen pitch. No new date has been set.
Benin midfielder Sessegnon tapped in from a corner on 52 minutes to make sure last season's losing finalists survived the first round involving top-flight clubs.
