France's triumph at the Burswood Dome sees them progress to Saturday's final, where they will face either the Czech Republic, Denmark or Bulgaria, with the outcome to be decided on Saturday.
World number nine Bartoli gave France a winning start with a 6-2 6-4 victory over Anabel Medina Garrigues, the world number 27.
Gasquet then gave France their second victory of the day by the same scoreline when he despatched Fernando Verdasco.
Depending on who progresses to the final, Bartoli will now face either Caroline Wozniacki or Petra Kvitova, the world number one and two respectively, or number 46 Tsvetana Pironkova.
It is the second time France have reached the final after Mary Pierce and Cedric Pioline went down to Karina Habsudova and Karol Kucera of the Czech Republic in 1998.
Elsewhere, Australia beat China 2-1 in the last tie for both nations at this year's event.
In the women's singles, Li Na made it three from three in her matches beating Australia's Jarmila Gajdosova 6-3 6-2.
Lleyton Hewitt then beat Wu Di 6-4 7-5 to force a mixed doubles which he won alongside Gajdosova for an overall victory.