US Open champion Alcaraz won 6-4, 7-6 (7/1) against his 74th-ranked opponent at home in Valencia, after Roberto Bautista Agut had earlier beaten Hong Seong Chan 6-1, 6-3.
After their surprise 2-1 loss to Canada on Friday, in which Alcaraz lost to Canadian Felix Auger-Aliassime, the Spaniards won in the singles against South Korea.
First in Group B ahead of Canada, the Spaniards will participate in the quarter-finals in November, in front of their home crowd in Malaga. Serbia have been eliminated.
In Hamburg, Germany took the top spot in Group C ahead of Australia after doubles pairing Kevin Krawietz and Tim Puetz's 6-4, 6-4 victory over Matthew Ebden and Max Purcell.
That gave the hosts a 2-1 victory.
In singles, Thanasi Kokkinakis beat Oscar Otte 7-6 (8/6), 6-1 to keep Australia in the tie after Germany's Jan-Lennard Struff eased past Purcell 6-1, 7-5.
Germany and Australia had already secured the top two spots in Group C with wins over France and Belgium earlier in the group stage.
In Bologna, Croatia advanced to the last eight thanks to Italy's 2-1 win over Sweden in Group A.
Croatia had started the competition with a heavy 3-0 loss to Italy before reacting against Sweden (2-1) and then Argentina (3-0).
The Croatian team took second place, qualifying behind Italy, while Sweden, third, were eliminated.
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