Cavani -- who has scored 18 times in 23 games for PSG since he joined from Napoli in the close season -- had given the hosts the lead in the first half only for former PSG striker and Turkey international Mevlut Erding to level in the second period to force extra time.
However, Cavani grabbed the winner two minutes from the end of extra time when he knocked home Jeremy Menez's low cross.
It was revenge for PSG as they had been knocked out by Saint-Etienne on penalties in the competition last season.
PSG's bitter historical rivals Marseille also progressed by the same scoreline against Toulouse which resulted in their interim coach Jose Anigo announcing he would still be in the role after the winter break as he celebrated his first win in charge.
Former Toulouse star Andre-Pierre Gignac grabbed Marseille's winner from the penalty spot in a competition that Anigo -- who stepped in after Elie Baup was sacked earlier this month -- said is low on his list of targets compared to the French Cup despite Marseille having almost made it their own in recent years having won it in 2010, 2011 and 2012.
"I am very happy with this result," Anigo said.
"But I know that first thing in the morning it will be time to sit down and assess how we approach this weekend's game with Bordeaux which will not be an easy one as they are the side playing with the most confidence at the moment."
Lyon, who have won the League Cup just once in 2001, twice took a two goal lead only to see Reims come back to trail by just a goal.
Goals by France international forward Bafetimbi Gomis and Alexandre Lacazette -- both scorers in Sunday's 2-2 draw with Marseille -- seemingly put Lyon in a comfortable position at 2-0 up by the hour mark.
Gomis nodded home at the far post from Lacazette's cross in the 58th minute while the latter extended their lead on the hour with a super right-footed shot after excellent lead-up work by playmaker Yoann Gourcuff.
Gourcuff restored their two goal advantage -- after Prince O'Niangue had made it 2-1 -- eight minutes from time with a cracking shot from long range having already gone close only to see his shot come back off the post.
However, Lyon's habit of letting leads slip throughout the season -- they led Marseille 2-0 last Sunday before being held to a draw -- looked as if it was going to happen again as Reims scored four minutes from the end of normal time through Floyd Ayite's penalty.