"It was very special to be back home after the (2023) World Cup triumph and the full crowd made it beautiful for us," Kolisi said after the clash of the top-ranked rugby nations.
"The support we have is very special, it carries us through, and our purpose is what drives us," he added after a match in which the Springboks led 13-8 at half-time.
"It was a big, tough game that delivered a high standard of rugby. It was important for us (to win) and say thank you to our supporters."
South Africa were playing at home for the first time since last July, when they defeated Argentina in the Rugby Championship in Johannesburg.
Since then, they have played 10 matches in Europe, with the highlight being the 12-11 victory over New Zealand in Paris last October to clinch back-to-back World Cup titles.
Reflecting on his performance, loose forward Kolisi said: "I can always get better (given) the opportunities I had. I did the best I could."
Looking ahead to the second and final Test of the series in Durban next Saturday, the captain admitted that "as a team we still have a lot to do".
Veteran Ireland skipper and loose forward Peter O'Mahony offered no excuses as a run of three straight victories over the Springboks from 2017 ended.
"It is always very difficult coming to South Africa and it was an unbelievable Test match. A couple of errors put us under pressure.
"South Africa attacked very well, their width was a bit of a change for them, but we fought well," added the forward making his 106th Test appearance.
"At this level there are always fine margins. We missed a few things on attack in the first half and they caught us out in attack on a couple of occasions."
Referring to a disallowed James Lowe try in the second half, O'Mahony said: "I am sure the referee would have got the disallowed try decision spot-on, but we will have a good look.
"I am sure there (were) some defensive decisions we could have made better."
In Durban, Ireland must win to avoid a fifth straight series defeat by the Springboks, whose lone loss in the 2023 World Cup campaign in France came at the pool stage against the Irish.
The countries have met in the Indian Ocean port only once before with South Africa winning 12-10 in 1981.
