The French will face the top-seeded Polish pairing of Agnieszka Radwanska and Grzegorz Panfil in Saturday's final.
France previously reached the title decider in Perth in 1998 and 2012.
Two years ago, the team of Richard Gasquet and Marion Bartoli lost the final to the Czech Republic pairing of Tomas Berdych and Petra Kvitova.
In 1998, Mary Pierce and Cedric Pioline were beaten by the Slovak Republic's Karina Habsudova and Karol Kucera.
Having won their first two group ties, the French simply needed to beat Spain's Anabel Medina Garrigues and Daniel Munoz-De la Nava to qualify for the final.
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Cornet said the problem was nothing more than early season tightness and, after Tsonga sealed the win for the French with a three-set win over Munoz-De la Nava, she took her place in the dead mixed doubles rubber.
The 23-year-old Cornet, who has lost both her matches on the WTA tour against Radwanska, was thrilled to have the chance to win France's first Hopman Cup title.
While other players have commented they are viewing the unique tournament as little more than Australian Open preparation, Cornet said she was in Perth to win.