The superstar had equalled Edinson Cavani's previous record mark of 200 goals with a brace in last week's win at Marseille and went one better by netting in stoppage time at the Parc des Princes.
PSG had earlier squandered a two-goal lead, with Lionel Messi putting them ahead and a Jaouen Hadjam own goal making it 2-0 inside 17 minutes before Nantes fought back to draw level.
Ludovic Blas pulled one back for the visitors and Ignatius Ganago equalised, but Mbappe set up Danilo Pereira to put PSG back in front on the hour mark and then claimed his record-breaking goal late on.
Mbappe reached the record in just 247 appearances since signing from Monaco aged 18 in 2017. Uruguay star Cavani needed 298 games, and seven years, to establish the previous record in 2020.