Fabien Barthez blocked Nicolas Marin's attempt and the decisive penalty was scored by Frederic Da Rocha, sole survivor of Nantes's 1-0 victory over Sedan in the 1999 final.
Sedan, bottom of Ligue 1, took the lead against the run of play in the 13th minute when Gregory Pujol, who started his career at Nantes, headed home from a Morgan Amalfitano cross from the left.
The Canaries, 19th in the league standings one point above Sedan, equalised two minutes later through Claudiu Keseru when the Romanian striker's low, 20-metre drive was deflected into the net by visiting defender Romain Sartre.
Nantes fight back to shock Paris Saint-Germain
Sedan then took control of the game but substitute Mansour Boutabout missed a chance for victory when he hit the woodwork with 14 minutes remaining.
Meanwhile Marseille cruised into the semi-finals with a 5-0 thrashing of third-division Vannes. L'OM, who have not won a trophy since winning the Champions League in 1993, went ahead in the second minute through Mamadou Niang.
Fellow striker Toifilou Maoulida doubled the tally from the spot after a handball by Vannes defender Pierre Talmont. Talmont then made the same mistake 16 minutes into the second half with France striker Djibril Cisse converting the penalty.
Maoulida netted his second goal five minutes later with a rising shot after being set up by playmaker Samir Nasri. Substitute Mickael Pagis rubbed salt into the wounds when the flicked the ball home in the last minute.
"It was a solid display, we just did what we had to," midfielder Lorick Cana said of the result.
On Wednesday, holders Paris St Germain travel to Sochaux and amateurs Montceau entertain RC Lens.
PSG heap misery on Marseille to keep pace with leaders
David at the double as Lille restore two-point lead in Ligue 1