Serbia, without Novak Djokovic, proved no match for Slovakia who surged into the group stage of the Davis Cup finals for the first time, with Germany and Finland among five other countries qualifying on Saturday.
Slovak doubles duo Lukas Klein and Igor Zelenay clinched the decisive point with a 7-6 (7/4), 6-3 win over Nikola Caci and Miomir Kecmanovic to take an unassailable 3-0 lead in the tie.
Lukas Pokorny made it 4-0 after Laslo Djere retired while trailing in the first set of the reverse singles on indoor clay at Kraljevo.
"Too many years we didn't qualify but finally we made it," said Zelenay.
Alex Molcan, who won Friday's second singles, added: "We knew it was going to be tough against Serbians in their home nation. I'm happy we were able to win two matches today."
Zelenay and Klein had recovered from a break in the first set, digging deep with 31 winners to see off 2010 champions Serbia.
On Friday, Klein got Slovakia off the mark with a 7-6 (7/2), 6-2 win over Kecmanovic, with Molcan rallying past Dusan Lajovic 4-6, 6-2, 6-0.
In Hungary, three-time winners Germany – missing Alexander Zverev due to illness – clinched the first two of Saturday's matches for an eventual 3-2 victory.
Doubles pairing Kevin Krawietz and Tim Puetz's 6-3, 7-6 (7/3) win over Fabian Marozsan and Mate Valkusz pulled the Germans ahead.
Jan-Lennard Struff sealed their ticket with a 6-3, 6-2 win over Valkusz to advance to September's group stage featuring 16 teams. Zsombor Piros won a dead fifth rubber against Krawietz.
Finland cruised to a 3-1 win over Portugal, while Brazil qualified for a first finals appearance as Thiago Monteiro defeated Elias Ymer in three sets to cement a 3-1 victory of their own.
DUTCH DELIGHT
The Netherlands won a thrilling contest against Switzerland, Botic van de Zandschulp edging Marc-Andrea Huesler 5-7, 7-6 (7/5), 6-3 in the decisive second reverse singles as the Dutch prevailed 3-2 in Groningen.
The Swiss led the tie 2-1 after Huesler and Leandro Riedi won Saturday's doubles, but Tallon Griekspoor kept the hosts alive by seeing off Riedi in two tie-break sets.
"We had to win and it was up to me to do it," said Griekspoor. "That is what you have to do as a team leader. I really didn't feel any tension - I really enjoyed it."
Canada, the 2022 champions, seized a 3-1 lead over South Korea to clinch victory on Saturday in Montreal when Gabriel Diallo defeated Hong Seong-chan 7-5, 4-6, 6-1.
Two-time winners Croatia are level at 1-1 with Belgium as are Argentina and Kazakhstan.
France hold a 2-0 lead against Taiwan, in one of five qualifiers set to conclude on Sunday. The Czech Republic have a similar advantage over Israel.
Defending champions Italy and last year's runners-up Australia, as well as Britain and Spain have advanced automatically to the finals.
On Friday, the United States advanced 4-0 with Austin Krajicek and Rajeev Ram sealing their spot with a doubles win against Ukraine.
Taylor Fritz then eased past Vladyslav Orlov 6-3, 6-4 to complete an American sweep.
"I'm proud of the guys," said US captain Bob Bryan of the record 32-time winners who have not lifted the trophy since 2007.