Julio Tavares came close to leveling things up when he was denied only by the palm of Remy Vercoutre, although the Ligue 2 side did manage to find a way through shortly after when Zakaria Diallo headed home.
Caen did not have to wait long to regain their advantage, however, taking the lead once more shortly before the half-time interval courtesy of Jordan Adeoti's strike into the top corner.
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With 10 minutes remaining Lois Diony found the target to rescue the second-tier side and take the game to extra time, just minutes after being introduced from the bench.
A dramatic evening was given a fitting end in the 105th minute when Florent Mollet found the space in the box to execute the perfect lobbed finish and send Dijon through to the last 16.