Stakusic, ranked 258th in the world, put Canada ahead by defeating No. 43 Martina Trevisan 7-5, 6-3 for the biggest win of her career, then Fernandez sealed the victory by beating Jasmine Paolini 6-2, 6-3 at La Cartuja Stadium.
The 20th-ranked Fernandez, runner-up in the 2021 US Open, was perfect for Canada in Seville with four victories, while the 18-year-old Stakusic entered the tournament without a win over a top-100 opponent but earned three of them while representing her nation in Spain.
"I'm so happy and honored that I could play this week," Stakusic, who did not play a WTA main draw match this year, said on court. "This has been the best week of my life."
Canada defeated 11-time champion Czech Republic in the semifinals, while four-time champion Italy advanced past Slovenia for its first final appearance since 2013.
The 12-team BJK Cup Finals offered a record total of $9.6 million in prize money, including $2.4 million to the champions, the same as the men's Davis Cup.
Canada, captained by Heidi El Tabakh, is the 13th nation to win the Billie Jean King Cup and the second new champion after Switzerland's triumph in Glasgow a year ago.
"I don't know what to say," El Tabakh said. "I'm so proud of this team. These girls are incredible. It's a dream come true."
The Canada men's team is the current Davis Cup champion.
The women's teams competed in four round-robin groups, with the winners advancing to the semifinals. The United States was eliminated by the Czech Republic in a group that also included title-holder Switzerland.
The biggest team competition in women's tennis started two days after the end of the WTA Finals in Cancun, Mexico, which featured the top eight players on the tour -- including winner Iga Swiatek.
The Associated Press and Reuters contributed to this report.