Bulka is on loan at Nice from PSG and stopped efforts from Leandro Paredes and teenager Xavi Simons in the shootout to end PSG's dominance in the cup, which they had won in six of the past seven seasons.
Nice will now host Marseille in the quarterfinals, after the draw was conducted earlier on Monday.
The home side laid siege to the Nice goal after an incident-free first half at the Parc des Princes and there were numerous chances in the second period for Julian Draxler, Mauro Icardi, Lionel Messi, Marco Verratti and Kylian Mbappe after he came on as a substitute.
But the closest PSG came was when defender Presnel Kimpembe had an effort cleared just short of the line as he almost steered a set piece into the Nice goal in the 71st minute.
The visitors, who are second in the Ligue 1 standings behind the Parisian club, had two counter-attack chances for Justin Kluivert blocked.
In next week's quarterfinal, the two fourth division giant killers Bergerac and Versailles were drawn against each other while the other two ties see a top-flight team host Ligue 2 opposition. Nantes are home to Bastia and Monaco take on Amiens.