PSG were then awarded a second spot kick late on following a Video Assistant Referee (VAR) review, but Walter Benitez saved Cavani's effort.
Thomas Tuchel's side have now won just one of their last seven games in all competitions, losing four and drawing two.
Despite the recent slump, PSG secured the Ligue 1 title last month and remain 17 points clear at the top of the table, while Nice are seventh on 52 points.
Patrick Vieira's side took the lead within a minute of the second half kicking off when Ganago latched onto a long ball from Dante and smashed an angled half-volley into the roof of the net.
Nice were inches away from a second just before the hour mark when the lively Youcef Atal skipped to the byline and cut a low cross to Allan Saint-Maximin on the edge of the six-yard box, but his finish came back off the crossbar.
The visitors were made to pay for their wastefulness less than a minute later when Patrick Burner brought down Angel Di Maria to concede a penalty which Neymar rolled in for his first league goal since January, following a lengthy injury lay-off.
The former Barcelona forward reached the landmark of 50 goals in just 57 games for the French club.
PSG had a golden chance to take the three points when they were awarded another penalty for Dante’s late challenge on Neymar, but Cavani’s tame shot was parried by Benitez.
(Reporting by Alasdair Mackenzie; Editing by Christian Radnedge)
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