Hosts Spain will take on last year's runners-up Croatia in the Davis Cup quarter-finals in Malaga in November.
The Netherlands and the United States, who both beat Great Britain in Group D of qualifying, face Australia and Italy in the last eight respectively.
Germany will play Canada in the other quarter-final match.
The Davis Cup knockout stage starts on 22 November at the Martin Carpena Arena with the final taking place five days later.
Spain, champions in 2019, knew they had to beat winless South Korea to advance in the final round of group ties on Sunday.
Straight-set wins for Roberto Bautista Agut and world number one Carlos Alcaraz in the singles saw them cruise to the last eight.
Alcaraz had suffered a 6-7 (3-7) 6-4 6-2 defeat by Canada's Felix Auger-Aliassime on Friday in what was his first match since winning the US Open.
However, the 19-year-old beat Kwon Soon-woo 6-4 7-6 (7-1) and Bautista Agut defeated Hong Seong-chan 6-1 6-3 to secure their place in the knockout stages.
Davis Cup organisers said they were "really proud" of this week's finals group stage despite numerous dead rubbers and matches finishing late.
On Thursday former world number one Andy Murray said playing tennis matches late at night does not look very "professional".
The Briton was still on court at nearly 01:00 BST on Thursday in Glasgow, playing doubles with Joe Salisbury in Great Britain's defeat by the United States.
Murray was again in action for GB against Kazakhstan on Sunday - a consolation tie for both sides after each started with two losses.
But Enric Rojas, chief executive of the International Tennis Federation's commercial partner Kosmos, said he felt the new format is "really working".
"I cannot avoid saying that we have faced a few challenges and issues, and they are under review," he said.
"We are really proud about the outcome of this week. I think that we have proven that the format works.
"Now with the four different cities, with the four groups, then going to the final eight to Malaga, it's a format for staying."
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