The forward was making his first club appearance of the new season after testing positive for coronavirus while on international duty with France.
Mbappe scored a penalty that he won himself and then saw his saved shot converted by Angel Di Maria on the rebound as two PSG goals towards the end of the first half proved pivotal.
Nice was not at full strength because of the impact of COVID-19 but Patrick Vieira's men were occasionally threatening until PSG captain Marquinhos headed in after the break to seal the win.
After Walter Benitez saved Di Maria's volley, the visitors lost Idrissa Gueye to injury midway through the half, with Ander Herrera replacing him shortly before a golden opportunity for Mbappe to open the scoring.
The France international looked sure to score after a wonderful passing move involving Mitchel Bakker and Mauro Icardi, only to fire over with a left-footed effort.
Presnel Kimpembe was somehow unable to get on the end of a Di Maria corner and later miscued another effort, before Mbappe atoned for his miss.
Seven minutes before the break, Mbappe confidently sent Benitez the wrong way from the penalty spot after a purposeful run into the box had seen him fouled by Khephren Thuram.
Herrera almost made it two but Andy Pelmard deflected his shot onto the post, before lively Nice winger Rony Lopes, who had earlier curled just wide, forced Navas into a key save after a great one-two with Amine Gouiri.
But PSG moved firmly into control one minute into first-half stoppage-time, Mbappe's magnificent run and shot forcing Benitez to parry, with Di Maria on hand to power in on the follow-up.
Navas was impressively at full stretch to make a fingertip save from Youcef Atal after the break, while Benitez had to rush out to deny Icardi with his legs at the other end, with the former Inter striker hitting the post from close range soon after.
It was game over after 66 minutes, though, Marquinhos heading in a superb in-swinging free-kick from Di Maria.
A second Mbappe goal was ruled out for offside, while Navas saved well from Evann Guessand late on to preserve PSG's clean sheet.