Otto Virtanen got Finland off to a solid start by outlasting Mackenzie McDonald 7-6(5) 1-6 7-6(7) before Emil Ruusuvuori sent Finland into the last eight from Group D with a hard-fought 7-6(1) 6-4 win over Tommy Paul in Split.
"It means a lot for the whole country and for tennis in Finland. It's a huge thing and once again we proved how strong this team is," Ruusuvuori said. "We're going to Malaga.
"I skipped the US Open as I was sick and didn't come here in really good shape. I struggled today physically and it was a bit tough in the beginning. But you just have to keep fighting. I was able to find a way in two long sets."
In Group C, already qualified Serbia lost 3-0 to the Czechs after world number one Novak Djokovic had opted to skip the singles matches having guided them to victory over Spain a day earlier.
Dusan Lajovic crashed to a 6-3 6-2 defeat by Jakub Mensik to leave Serbia in a spot of bother in Valencia.
Laslo Djere, who has been in good form during the Finals group stage, overcame a back problem and dug deep to save four match points in his clash with Jiri Lehecka but lost 7-6(7) 7-5 as the Czechs prevailed.
They then finished the job in the doubles, becoming the first country to ever win all nine of their matches.
"I'm pleased with the win, it's as simple as that. Laslo is a great guy and even better player so for me to play against him on this court was such a big challenge," Lehecka said after ensuring top spot in the group for his nation.
"He plays some incredible shots and he never gives up. Even when I was up 4-1 in the second set, he was still there waiting for his moments. I must have added something to my level and it went well."
The teams will next head to Malaga for the knockout rounds that take place from November 21st-26th.
They will be joined by Australia, after last year's runners-up secured a 3-0 win over Switzerland in Group B in Manchester.
Thanasi Kokkinakis beat Dominic Stricker 6-3 7-5 and Alex de Minaur downed Marc-Andrea Huesler 6-4 6-3 before the pair of Max Purcell and Matthew Ebden added some gloss to the victory with a dominant display in the doubles rubber.
"The first two boys did so good to put us up 2-0 and made us feel extra confident by taking the pressure off. We knew that we still had to win this match," Ebden said after the duo scored a 6-2 6-4 win over Huesler and Stricker.
"Huge team effort all week from everyone."
Canada's world number 200 Alexis Galarneau continued his superb run this week as he got past Chile's Alejandro Tabilo 6-3 7-6(5), with the match win ensuring that the defending champions will reach the next round of the competition.
Nicolas Jarry levelled the Group A tie in Bologna by beating Gabriel Diallo 6-4 6-4 before Canada clinched the deciding doubles rubber.
