Licata came off the bench in the 79th minute and made an immediate impact as he headed home from a Francois Modesto cross six minutes from time.
Monaco, whose last home win against Nice was in 1996, will next travel to Ligue 2 AC Ajaccio for a quarter-final spot.
Earlier, Norwegian striker Thorstein Helstad struck eight minutes into extra time to give Le Mans a 1-0 win at Le Havre in another all-Ligue 1 clash.
Helstad poked the ball into an empty net after being set up by Ivory Coast forward Gervinho and Le Havre, who are bottom of the Ligue 1 table, proved too clumsy to come back.
Grenoble survived a scare to see off sixth division side Grande-Synthe 3-1 after going 1-0 down in the 12th minute through a Florian Franchois goal.
The Ligue 1 team then stepped up a gear and prevailed courtesy of goals from Nassim Akrour, Larsen Toure and Bouchaib El-Moubarki.
Last year's finalists Paris St Germain went through with a 3-0 win at fourth division Ajaccio GFCO with a Mateja Kezman double and a Sammy Traore goal.
The capital side will next travel to third division Rodez.
Olympique Lyon take on Olympique Marseille on Wednesday in the final last 32 match, with the winner travelling to Lille in the next round.
