The game appeared to be heading for a stalemate until Delort met Calvin Stengs' pinpoint cross with aplomb, as PSG suffered their second consecutive road defeat.
The situation could have been even worse for PSG had Mauro Icardi's foul on Jordan Lotomba not been ruled beyond the area.
Nice's victory comes after a triumph against PSG in the French Cup in January.
Nice's Justin Kluivert missed a chance moments before Delort's goal when he was clear through on goal but could only manage a weak effort at Keylor Navas.
Nice moved into second place, ahead of Marseille, after Angel di Maria squandered an opportunity after being played in by Neymar.
With a front three of Di Maria, Neymar, and Lionel Messi, the visitors only managed two shots on goal.
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