PSG asserted the early pressure, but Leonardo Paredes' shot off a free-kick narrowly missed the intended target after nine minutes.
Moussa Diaby followed suit with a similar opportunity, however the French winger failed to cap off Christian Nkunku’s cross on 13 minutes.
Before halftime, Villefranche launched their first sight on goal through Nicolas Burel’s close-range shot that shaved the left-hand post.
In the second half, the hosts went close once again when Celin Lemb created space near the box for a long-range shot that grazed the top of Alphonse Areola’s crossbar.
Whilst struggling to unlock the unfancied side, PSG’s idea to changing all that saw Kylian Mbappe coming off the bench for Di Maria.
With time running out and extra-time looming, Mbappe stood broadly over a direct free-kick but the teen sensation failed to get the effort on target as it remained goalless after regulation time.
The opener arrived in the 102nd minute as Julia Draxler showed great composure on the ball to tuck home Cavani’s pass.
Twelve minutes later, Mbappe’s cleverness made way for the second as Diaby cushioned home the former’s perfectly weighted delivery.
The match wouldn’t be complete without a certain Cavani getting in on the act, as he plugged in the third a minute before the match to confirm a quarter-final spot for PSG.