Kluivert netted with a firm downward header for the first, then a powerful looping shot over the goalkeeper from just outside the penalty area for the second. For good measure he set up striker Andy Delort for the fourth goal with a curling pass from the right flank.
Kluivert is on a season-long loan from Italian club Roma. The 22-year-old has played for the Netherlands but not scored. His father Patrick Kluivert netted 40 times for the Oranje and scored 122 goals in 257 games for Barcelona from 1998-2004.
Scoring runs in the family, with Patrick's father Kenneth Kluivert notching a hefty 366 goals in 345 games for Surinam team Robinhood in Paramaribo.
Marseille scored first from a rapid attack in the third minute led by midfielder Matteo Guendouzi.
Poland striker Arkadiusz Milik's shot was blocked and the rebound found Turkey winger Cengiz Under, whose shot was stopped but hit the back of defender Melvin Bard and rolled in.
Nice forward Amine Gouiri equalised in the 10th after a fine solo run.
Gouiri was involved in the second, laying the ball off to Delort on the right and his cross was met by Kluivert's head.
Kluivert struck again in the 49th with a powerful shot from the right into the left corner of the net.
Nice won the last of their three French Cups in 1997.
Earlier, Versailles knocked out fellow fourth-tier side Bergerac on penalty kicks to reach the last four.
Versailles scored through midfielder Inza Diarrassouba's 14th-minute goal, but Axel Tressens equalised after receiving an astute pass from Christian Gyeboaho in the 90th.
The handful of fans who traveled some 580 kilometres down from near Paris celebrated jubilantly with the players after Versailles won 5-4 on penalties to join last year's runner-up Monaco in the last four.
Nantes face second-tier Bastia at home on Thursday.
Nantes were once a powerhouse in French soccer and celebrated winning their third French Cup in 2000 and their eighth league title the following year but have won nothing since 2001.
Coach Antoine Kombouare, a former Nantes defender, has brought much-needed stability back to the club.
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