Ferran Torres scored the only goal of the final in the 106th minute, securing Spain's second World Cup title and adding to the European Championship crown they won under De la Fuente in 2024.
The Spain coach struggled to hide his emotions after the final whistle, praising the character, unity and quality of his squad.
"I feel especially proud of this generation of footballers who have grown with this idea of being a team and giving us an example of a group, of a family," De la Fuente said.
"They are players with exceptional talent. For me, it has been an honour to have accompanied them on this journey."
The 65-year-old reflected on what the squad has achieved during his time in charge.
"I'm very emotional when I look back. We have won everything, and that is wonderful," he said.
"They are a generation of footballers who are a source of pride for Spain. Together, we are stronger."
Spain dominated long periods of the final but had to wait until extra time to find the breakthrough. De la Fuente felt his side had done enough to settle the contest earlier but credited Argentina goalkeeper Emiliano Martinez for keeping the holders in the game with a series of outstanding saves.
"I think the game should have been decided earlier. Dibu's great saves prevented us from winning sooner, but it is a World Cup final and, even with ten players at the end, you have to suffer," he said.
Argentina's task became even harder when Enzo Fernandez was sent off for a second yellow card in the 93rd minute, leaving the defending champions to play almost all of extra time with 10 men.
Spain eventually made their advantage count when Torres struck the decisive goal, capping a memorable night and confirming De la Fuente's team as world champions after another remarkable tournament.