Unai Emery s side advance to the last 32 of a tournament in which they are the two-time defending champions ensuring their Corsican rivals remain winless at the Parc des Princes.
Making his competitive debut 23-year-old winger Draxler completed the rout scoring the final goal three minutes from the whistle.
The German World Cup-winner completed his move this month from Wolfsburg for a reported fee of up to 42 million euros ($45 million).
PSG captain Thiago Silva opened the goal fest with a header on 30 minutes with Adrien Rabiot firing in the second from 25 metres out just before the break.
It was all one-way traffic in the second half with goals Christopher Nkunku, Thiago Motta, Angel Di Maria and a Lucas penalty humiliating the visitors before Draxler struck.
Emery s side, who went into the winter break in third place in Ligue 1, five points behind leaders Nice, are looking to complete a clean sweep of the domestic trophies in France for a third season in succession.
The Parisians were however hit when Brazilian captain Silva picked up a thigh injury in a challenge with goalkeeper Kevin Traff minutes before the break.
The 32-year-old was treated pitch side for several minutes before limping off.
Five other Ligue 1 sides also competing on Saturday advanced.
Lille ended the adventure of minnows AS Excelsior — a sixth-tier side from the Indian Ocean island of Reunion — 4-1, as Guingamp beat Le Havre 2-1.
Monaco on Friday advanced past second division AC Ajaccio 2-1, largely thanks to a goal from star forward Falcao.
On Sunday, Marseille travel to Toulouse with Lyon hosting Montpellier.