Poland - led by world No.1 Iga Swiatek - Australia, the United States, Japan, Slovakia, Romania and Germany were the other countries that advanced to the finals, which will be held in Seville, Spain, in November.
The eight nations from qualifying join defending champions Canada, 2023 runners-up Italy, hosts Spain and wild-card selection Czech Republic in the 12-team finals field.
Raducanu, the 2021 US Open champion who has since struggled with poor form and a string of injuries, also put together a comeback from a set down in her opening singles match on Friday against Caroline Garcia on an indoor clay court in Le Portel, France.
THE MOMENT ✨— wta (@WTA) @EmmaRaducanu | #BJKCup pic.twitter.com/25m6tQZXqeApril 13, 2024
This was a rematch from last year in qualifying, when France won; this time, the countries were tied at 1-1 after Friday's matches.
Katie Boulter beat Clara Burel 7-5 6-0 to put Britain ahead 2-1 on Saturday, before Raducanu - who had surgery on both wrists and an ankle in 2023 - came through in a contest that lasted nearly three hours.
She led Parry 5-2 in the third set and had two match points while serving for the win at 5-3, but needed the tiebreaker to close things out.
"To be able to put two matches like that back to back - three sets and on clay, against really tough opponents - it's just a testament to the work we've been doing," said Raducanu, who has been as high as No.10 in the WTA rankings and is currently ranked 302nd.
"I knew it would pay off, eventually. I'm very pleased that, this weekend, it was able to show."
Britain has never won the international team competition for women.
Australia went 3-0 up over Mexico in Brisbane thanks to 18-year-old Taylah Preston's 6-1 6-1 victory over Marcela Zacarias on Saturday.
The US moved on when Jessica Pegula put the hosts ahead 3-0 against Belgium by beating Hanne Vandewinkel 6-2 6-0 in Orlando, Florida.
The Americans have won a record 18 titles in the event that used to be known as the Fed Cup, but their last came in 2017.
Swiatek clinched a berth for Poland by defeating Switzerland's Celine Naef 6-4 6-3 on Saturday for a 3-0 lead.
Japan went ahead 3-0 against Kazakhstan without needing four-time grand slam champion Naomi Osaka to play a second match in Tokyo.
Slovakia moved into the finals when 16-year-old Renata Jamrichova defeated Veronika Erjavec 6-2 6-0 in Bratislava to give the hosts a 3-0 edge over visiting Slovenia.
After losing the first two matches, Romania rallied to beat Ukraine 3-2 in Florida.
Ana Bogdan and Jaqueline Cristian won singles matches to tie it, then teamed up to beat Ukrainian twins Lyudmyla and Nadiia Kichenok 6-2 7-6.
Germany beat Brazil 3-1.
