The Boks had led 12-6 at half-time in Paris, Handre Pollard kicking four penalties to two from Richie Mo'unga.
The sole score of the second half was a Beauden Barrett try for a New Zealand team reduced to 14 after Sam Cane was red-carded for a high tackle after 34 minutes.
"There are no ways I can explain it," said Kolisi, who also received a yellow card in the second half for a high hit.
"I want to give credit to the All Blacks. They took us to the end, they took us to a dark place.
"It shows what kind of team they are, to fight with a man down from early in the game. They put us under so much pressure."
Kolisi added: "Credit to my boys too for the fight. I am just grateful we could pull it off.
"People who are not from South Africa don't understand what it means for our country. It is not just about the game. Our country goes through such a lot."
He added: "I want to tell the people of South Africa 'thank you so much'. This team just shows what you can do.
"As soon as we work together, all is possible, no matter in what sphere -- in the field, in offices, it shows what we can do. I am grateful for this team, I am so proud of it."
The Bok skipper admitted that losing hooker Bongi Mbonambi in just the third minute had been a blow to his team, but they had re-organised and battled through.
"We lost our hooker in the early part of the game and we had to adjust to that. They put a lot of pressure on our lineouts but somehow we found a way," he said.
Coach Jacques Nienaber said the record fourth title after previous successes in 1995, 2007 and 2019, had been long in the planning.
"We came a long way with the players," said Nienaber, who took over from Rassie Erasmus as head coach afer the latter had led the Boks to glory in Japan four years ago.
"We always planned for this World Cup, since 2018. Massive credit to our fans. They were special from South Africa, 62 million people, the videos we had to play from them. It was unbelievable."