Nice, into the last four for the first time in 11 years, will visit fourth division Versailles in the semi-final after the draw was conducted at the end of Wednesday's match.
Kluivert, 22, headed home at the end of a quick counter-attack in the 29th minute to make it 2-1 before following that up with a right-footed scorcher four minutes into the second half as Nice went two goals clear.
The Dutch international also provided the pass for Andy Delort's 61st minute back-post, half-volley, which put the result beyond doubt.
Earlier on Wednesday, Versailles overcame Bergerac away 5-4 on penalties in a clash of two fourth-tier teams in the last eight. The match ended 1-1.