Thomas Tuchel, who watched the conclusion from the stands after being sent off, selected a strong line-up for the tie, which was moved to Lyon's Groupama Stadium.
Kylian Mbappe and Cavani were pressed into action from the bench and the Uruguay star set up Draxler in the 102nd minute.
Moussa Diaby then enjoyed a similarly simple finish and Cavani got in on the act himself in the final seconds.
Teenage winger Diaby should have opened the scoring in the 13th minute but skewed wide after Christopher Nkunku prised Villefranche open on the break.
Angel Di Maria fired into the side-netting following a clever backheel from Eric Maxim Choupo-Moting but Villefranche grew into the contest and Nicolas Burel and Ali M'Madi were narrowly off target with speculative strikes.
Celin Lemb swivelled to fire just over from the edge of the area as the hosts continued to play with confidence early in the second half.
Tuchel was cutting an increasingly irritated figure in the touchline and, after he brought Mbappe on for Di Maria with an hour gone, he was sent to the stands for persistent complaints to the officials.
Cavani joined Mbappe in place of Choupo-Moting, forcing Antoine Philippon into a stunning save from his 83rd-minute free-kick.
Villefranche's goalkeeper was then out sharply to thwart Mbappe during the first half of extra time before holding a Cavani header on the line.
The increased pressure told in the when Marquinhos found Cavani with a scooped pass and he was able to drive in a low ball from the right. Draxler was left with the formality of scoring and averting huge embarrassment.
Alain Pochat's team pushed bravely after half-time in the additional period but that left space for Mbappe to stream clear twice and give Diaby and Cavani tap-ins, both goals coming in the final seven minutes.
What does it mean? Cups become a slog for specialists PSG
Heading into this season, PSG had won the previous four Coupes de France and past five Coupes de la Ligue. But Guingamp put them out of the latter competition on penalties before this unexpected ordeal. Tuchel was in a foul mood for as long as he was allowed to remain on the sidelines but bigger prizes are the priority. The reality is, this near-blemish will be quickly forgotten if he can plot a path to Champions League glory.
Paredes an assured presence
On his full debut, Argentina midfielder Leandro Paredes did not have a great deal to work with in between a skittish defence and ineffective attack. But the new signing displayed a pleasing range of passing on his full debut and kept things ticking smoothly enough, suggesting he will be a shrewd acquisition with that Champions League showdown against Manchester United approaching. The 24-year-old also went close with a couple of free-kicks during the first half.
Unexpected night of toil for Bernat
Juan Bernat might have expected to make way for 19-year-old Stanley N'Soki at left-back before Tuchel named an imposing XI. The Spaniard is unlikely to thank his coach for this experience, however, as he was sloppy in possession and frequently caught out defensively as Villefranche targeted his flank.
PSG host Bordeaux as they look for a return to winning ways in Ligue 1 on Saturday, while Villefranche face Dunkerque in a key relegation clash.