Despite the absence of five PSG players including superstar Lionel Messi and Danilo after positive tests for Covid-19, Mauricio Pochettino's side were never worried by the Brittany minnows.
Defender Presnel Kimpembe, wearing the captain's armband, headed in after 28 minutes just after the match had been interrupted for a minute after a drone flew over the Rabine Stadium in western France.
Kimpembe then set up Mbappe for the second just before the hour mark with the French forward completing a hat-trick in an 18-minute spell.
Mbappe added a second after 71 minutes after good work from Marco Verratti and Xavi Simons adding a third six minutes later for his 150th goal in a PSG jersey.
Vannes' Cup dream was over with the Brittany side also missing out on their hoped for bonanza after the French government's crowd restrictions with 5,000 spectators only permitted at outdoor events.
Vannes had been hoping for a 9,600 sell-out and according to their president Maxime Ray are set to miss out on revenue of 250,000 euros ($282,000).
"We are missing out on what is a fortune to us," he said.
Covid could also overshadow Tuesday's Lens versus Lille Cup clash with seven players from the visiting team confirmed positive.
"We had four Covid cases which has now become seven," said Lille coach Jocelyn Gourvennec, without naming the players.
"We're going to re-test all the players tomorrow (Tuesday) morning."
Monday's drone incident was another in a series of problems French football have had since the start of the 2021/22 season.
Sunday's Cup match between fourth-division Jura Sud and Saint-Etienne, in Louhans, was interrupted for about 20 minutes due to smoke and fireworks launched by supporters of the Greens.
Jura Sud goalkeeper Cedric Mensah described his shock on Monday after crowd trouble "cut the legs" of his team in a 4-1 French Cup defeat to Ligue 1 outfit Saint-Etienne.
"I heard a powerful bang and felt a pain in my right ear. I had the impression that a bomb had exploded and felt the pitch vibrate," he said.
"The doctor explained to me that if it had exploded closer, I could have pierced my eardrum. At the time, it really shook me."
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