The All Blacks scored two tries to one, gaining a significant edge when scoring 17 points in five minutes midway through the first half.
Captain Kieran Read said, "It was a heck of a Test match. It was the full 80 minutes right to the end there that we had to work and it's just fortunate some times it just happens in those couple of moments and we managed to take them and it was probably the only difference in the game.
"We had to defend a little bit obviously early and when you don't have the ball you can't do too much so when we did get opportunities obviously we want to try and speed the game up. It was very tough out there though, very tough conditions," he said.
Coach Steve Hansen was delighted with the win. He said, "Today was a big game for both sides and fortunately for us we came out on top but it was another titanic struggle between New Zealand and South Africa.
"They [South Africa] were never out of it, they are always a team that will come back. They're a little bit like ourselves, they don't die, they stay in the fight and they took a good opportunity and scored," he said.
While there were 10 days before their next game, against Canada, there would be a chance for those who played South Africa to recover while those who hadn't played would be keen to get some game time.
"We'll use the next couple of days wisely and then get back into work," he said.
South Africa coach Rassie Erasmus said, "They obviously deserved to win, two tries to one and a penalty count that was 10 against us and only two against them. That's great discipline by them and it showed on the scoreboard."
The Springboks conceded 17 points in five minutes, one of those from trying something funny in their own half, he said.
But after halftime they had clawed their way back.
"And then we couldn't convert when we were inside the 22m and if you concede 10 penalties against them they will kick them all over and we couldn't," he said.
While no side had won the World Cup when losing a pool game Erasmus said that would be a challenge the side would have to look to achieve.
"We can't moan about this performance, they really played well, I think the referee [Jerome Garces] was good and we just have to work really hard," he said.
First five-eighths Richie Mo'unga told Spark Sport his first World Cup game had lived up to its hype and it was a great Test match.
While the toughest game of the pool play was behind them, the All Blacks would be keeping their feet on the ground. They would celebrate the win but it was important they recovered well from what had been a tough game physically.
Fullback Beauden Barrett said the win had been huge as they wanted to start the tournament on a high note.
"There was no better way to do it tonight than against the Springboks so that's probably the bench mark for us and obviously there's a few points left out there but there's always going to be that against a quality team. We knew they'd come for us and they had their moments too," he said.
"There were so many close moments, some where it [the rush defence] came off but we knew that was coming. We knew our skill sets had to be at high levels to make the most of those opportunities.
"We probably didn't take as many of those as we wished we had but that's Test match footy and it comes down to small margins and being able to take as many of those opportunities as possible," he said.