Alcaraz, who reached the top of the rankings on the back of his triumph at the US Open, survived a tight second set to topple Kwon 6-4 7-6(1) after Roberto Bautista Agut had given Spain the lead by dispatching Hong Seong-chan 6-1 6-3.
"Davis Cup is different. You are playing with your team, represent your country. You don't play just for yourself. You're playing for everyone in Spain, you know," Alcaraz, who converted all three of his break points, told reporters.
"It is a different atmosphere, different air you breathe, totally different from the other tournaments."
Six-times winners Spain will meet Croatia for a place in the semi-finals.
In Group A action in Bologna, Berrettini faced little trouble in seeing off Elias Ymer 6-4 6-4 before Mikael Ymer levelled the tie for the visitors with an upset 6-4 3-6 6-3 win over in-form Jannik Sinner.
The veteran duo of Simone Bolelli and Fabio Fognini settled the tie in Italy's favour with a dominant 7-6(2) 6-2 win against Andre Goransson and Dragos Nicolae Madaras in the doubles decider and set up a last-eight clash with the United States.
"I'm feeling happy. Obviously the main goal was to qualify for Malaga. We did it after the first two matches," said world number 15 Berrettini.
"In the locker room today we were talking. We said the job wasn't done yet. We still want to make a statement for ourselves... we give everything, especially when we play in front of so many people that they're here to support us."
Tim Puetz and Kevin Krawietz led Germany to top spot in Group C when they stunned Wimbledon champions Matthew Ebden and Max Purcell of Australia 6-4 6-4 after Jan-Lennard Struff and Thanasi Kokkinakis had won their respective singles rubbers.
Germany will take on Canada in the quarter-finals with Australia going up against the Netherlands. The knockout rounds will be played in Malaga from Nov. 22-27.
