Bernard Casoni's charges had surrendered a 2-0 goal lead at the Stade de l'Abbe Deschamps on Thursday, but progressed as Lynel Kitambala's 121st-minute strike wrapped up a 3-2 victory.
Jamel Ait Ben Idir put the hosts ahead on 34 minutes, as Auxerre led 1-0 at the break.
The tie looked to be a formality when Paul-Georges Ntep de Madiba doubled the advantage on 64 minutes, but Dijon halved the arrears just two minutes later through Florin Berenguer.
Dijon hit level with just eight minutes remaining when forward Julio Tavares ensured extra-time.
The match was seemingly heading for penalties with neither side looking likely to find a breakthrough in the additional time, until Kitambala's dramatic winner at the death ensured the four-time winners would feature in the last eight.
Auxerre will be joined in the quarter-finals by Cannes who defeated Plabennec 1-0 at the Stade Pierre de Coubertin.
The tight encounter was settled by Belkacem Zobiri's winner in the 28th minute.
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