Thomas Darracq's team sealed their progress with a strong second-half showing despite being frustrated by an Italy side that defended with determination in their first appearance in the competition's knockout rounds.
Joanna Grisez nabbed a hat-trick for the French, opening the scoring with barely two minutes on the clock before completing a trio of tries with two scores late in the second half.
"We are so happy for this game, to play a semi-final in Eden Park next week," French centre Gabrielle Vernier said.
"I'm so proud of the girls. It's been a tough few weeks and today we proved that we are one of the best teams.
"It's amazing, a lot of us haven't played a World Cup before. It's a dream come true."
France made the perfect start when Emilie Boulard scooped up possession on the edge of her own 22 before bursting through the Italian defensive line and offloading to Grisez to the right, leaving the winger to sprint between the posts.
Michela Sillari's penalty briefly reduced the deficit while France's Caroline Drouin slotted between the posts after Maria Magatti was shown a yellow card late in the first half to give the French a boost going into the interval.
Charlotte Escudero and Boulard both had tries ruled out before the French were awarded a penalty try in the 61st minute as Silvia Turani was yellow-carded for impeding the scrum with France driving towards the line.
The loss of Turani saw the Italian resistance collapse, with Laure Touye adding another try from the back of a rolling maul in the 64th minute before Grisez added two more in the final 12 minutes.
"We are happy to have made the quarter-finals, but we need to start from this and improve in the next years so we can get to the semis at the next World Cup," Italy centre Sillari said.
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