Goalkeeper Clement Petrel made a couple of smart saves before PSG central defender Presnel Kimpembe stooped to score a powerful header from a Nuno Mendes corner in the 28th minute.
Mbappe was kept quiet by the Vannes defense in the first half on a rainy night in the Brittany region of northwestern France. But he made it 2-0 in the 59th when he latched onto Kimpembe's long pass from defense to sprint clear and finish with a low shot that looped in.
His next goal was far more emphatic as he arrowed a shot high into the net from 20 meters, after good work from midfielder Marco Verratti allowed 18-year-old Dutch midfielder Xavi Simons to send Mbappe through in the 71st.
His third goal was from close range after swapping passes with substitute Eric Ebimbe in the 77th. It marked his 18th goal in 25 games overall. PSG coach Mauricio Pochettino also picked 18-year-old French midfielder Edouard Michut, who impressed with his technical ability albeit against modest opposition.
PSG were without Lionel Messi, isolating at home in Argentina after being among four players who tested positive for the coronavirus during a brief winter break. Pochettino was uncertain if Messi will recover in time to play at Lyon in the league on Sunday.
Lens host Lille in a northern derby on Tuesday for the final spot in the last 16.