Mbappe put the defending champion ahead shortly before the midway point of the first half with his sixth league goal of the campaign, before Nantes wing-back Fabio received his marching orders for a high challenge.
The France international grabbed his second early in the second half, before Nuno Mendes added a third as Christophe Galtier's men wrapped up maximum points with the minimum of fuss.
The result meant PSG returned to the top of Ligue 1 after Marseille had briefly usurped them earlier in the day with victory over Auxerre, while Nantes remained in 10th.
Samuel Moutoussamy fired over for Nantes early on, while Danilo Pereira and Marquinhos went close at the other end as the visitors started to flex their muscles.
Mbappe then deservedly put PSG ahead in the 18th minute with a whipped effort past Alban Lafont after being played in by Lionel Messi at the end of an incisive counter-attacking move.
Nantes' task became significantly harder six minutes later when Fabio received a straight red card for an ugly lunge on Vitinha.
The game was effectively put out of their reach in the 54th minute when Mbappe prodded home from six yards after being picked out by Messi, the goal eventually given after a long VAR review for a possible offside.
Substitute Neymar inexplicably hit the post with the goal at his mercy, but his blushes were spared when Mendes fired in the rebound for his first PSG goal.