Spain, who last won the Davis Cup in 2019, had to beat South Korea to advance from the final round of group ties on Sunday.
They did so with a minimum of fuss, wrapping up the result after straight-set wins for Roberto Bautista Agut and world number one Carlos Alcaraz in the singles.
Alcaraz went down in three sets to Canada's Felix Auger-Aliassime on Friday, his first match since winning the US Open, but bounced back in fine style on Sunday, beating Kwon Soon-woo 6-4 7-6 (7-1).
Bautista Agut defeated Hong Seong-chan 6-1 6-3.
Also advancing to the last eight were the Netherlands and the United States, who will face Australia and Italy respectively.
The other quarter-final match-up will see Germany take on Canada.
The Davis Cup knockout stage starts on 22 November at the Martin Carpena Arena in Malaga.
Not everyone was pleased with the new condensed format, which resulted in several very late finishes and dead rubber matches, but Davis Cup organisers hailed the week a success.
"I cannot avoid saying that we have faced a few challenges and issues, and they are under review," said Enric Rojas, chief executive of the International Tennis Federation's commercial partner Kosmos.
"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."
Italy v USA
Germany v Canada
Spain v Croatia
Netherlands v Australia