Down 17-10 at halftime, the hosts charged into the lead through tries to Ruby Tui and Theresa Fitzpatrick and were well placed as France went a woman down with a yellow card for loose forward Safi N'Diaye for another high tackle in the 70th minute.
France number eight Romane Menager scored a brace of tries in a powerhouse display, her second pulling the French within a point with 14 minutes left.
But a yellow card for loose forward Safi N'Diaye for a high tackle proved costly for France before Drouin's late miss sealed their fate.
"It was just an unbelievable ending, really," New Zealand coach Wayne Smith said.
"I thought we were pretty jittery at the start ... I thought it was gone, but it wasn't gone. It's there and we've got a chance next week."
France have never made it to a World Cup title match and will rue a semi-final that got away from them after they did the hard work early.
New Zealand saw off a fierce, 17-phase assault on their try-line in the opening minutes but the pressure finally told as Menager crashed over at the left post in the 23rd minute.
The French defence was brutal and inside centre Gabrielle Vernier was lucky not to draw a card after lifting winger Tui in a tackle and dumping her on her back.
France saved two tries before finally conceding after Drouin kicked the ball out on the full in a rare mistake, giving New Zealand a scrum in front of the posts five minutes from the break.
Flyhalf Ruahei Demant flung the ball wide to Stacey Fluhler who streamed over near the left corner at pace to fire up the crowd.
Fullback Renee Holmes's conversion levelled the match at 10-10 but the relentless French came again, bashing away at the New Zealand line until Vernier burst over by the left post on the stroke of halftime.
FRENCH AGONY
The match went up a notch as Holmes sent the French defence scrambling with a deft chip forward four minutes after the re-start.
A human lightning bolt, Tui won a sprint against Emilie Boulard, toed the ball out of the fullback's grasp and chased it down for a try just before the dead-ball line.
Holmes missed the conversion, though, and then shanked a long-range penalty a few minutes later.
It scarcely mattered though as the Black Ferns' running game clicked into top gear.
A big carry by lock Maiakawanakaulani Roos drove the hosts two metres from their posts and Fitzpatrick barged over to snatch the lead for New Zealand near the hour-mark.
Demant slotted the conversion and a penalty to stretch the lead to eight points but Menager set up a grandstand finish as she pounded through a nest of Black Ferns for her second try.
Further drama was to come as Taumata crashed into hooker Agathe Sochat, the high tackle, spotted by the television match official, directly in front of the posts.
From about 30 metres out, Drouin's shot went well wide, leaving France in agony as Black Ferns scrumhalf Kendra Cocksedge booted the ball out and home fans rejoiced.
