Clara Burel put France on course for the 3-0 victory they needed to advance from Group A with a three-set victory over Ekaterina Alexandrova, but Anastasia Pavlyuchenkova outlasted Alize Cornet in the second rubber to ensure a new champion will be crowned.
Pavlyuchenkova saved six of 10 break points and racked up an impressive 34 winners en route to a crucial 5-7 6-4 6-2 success against Cornet.
"I'm so, so happy and so proud of myself because I think it was an incredible match. It was very good tennis but more importantly I think it was an amazing fighting spirit from both of us," said Pavlyuchenkova.
"We were like two tigers. Nobody wanted to give the other anything and I think that's what matters. That's the sport we're playing."
Veronika Kudermetova and Liudmila Samsonova comfortably defeated Cornet and Burel in the doubles to consign the reigning champions to a 2-1 defeat.
The RTF will go up against the USA in the first semi-final, with the Americans comfortably seeing off Spain in the singles rubbers.
Sloane Stephens bested Nuria Parrizas-Diaz 6-4 6-4 before Danielle Collins swept Sara Sorribes Tormo aside 6-1 6-0 in under an hour.
Collins won 83 per cent of points behind her first serve and forced Sorribes Tormo into 24 errors during a resounding victory.
Spain avoided a 3-0 defeat, though, with Aliona Bolsova and Rebeka Masarova beating Caroline Dolehide and CoCo Vandeweghe 6-3 6-4 in the doubles rubber.