Mladenovic and Garcia then teamed up to beat Bertens and Richel Hogenkamp, coming from a set down to win 4-6 6-3 6-3 in a nail-biting climax to the weekend.
France have reached their first final since 2005 where they will be up against the Czech Republic, who also needed a final doubles rubber to win in Switzerland.
The Netherlands were unbeaten since 2013 despite a team full of lower-ranked players, beating Australia last year to reach the World Group and stunning Russia in the first round.
Karolina Pliskova and Lucie Hradecka booked the Czech Republic's place in a fifth Fed Cup final when they beat Swiss duo Martina Hingis and Viktorija Golubic 6-2 6-2.
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Golubic, ranked 129 in the world, had kept the Swiss in the hunt with two singles wins.
Four-times champions Italy were relegated from the World Group for the first time in 18 years after losing 4-0 in a playoff with Spain who return to the elite after a two-year absence.
Belarus caused an upset by beating Russia 3-1 in their World Group playoff, meaning the Russians are also relegated from the top group for the first time in 18 years.
(Reporting by Martyn Herman; Editing by Ken Ferris)
