In a tight game at Marseille's Stade Velodrome, two Manie Libbok penalties to one from Finn Russell made it just 6-3 at halftime.
Momentum swung after Pieter-Steph du Toit and Kurt-Lee Arendse crossed for a try apiece between the 47th and 50th minutes, the Boks effectively shutting down any response from the Scots after ensuring dominance on the scoreboard.
"It could have been a slippery one, Scotland are a good side and are not No 5 in the world for nothing," said Nienaber, who took over as head coach from Rassie Erasmus, who moved to director of rugby after leading the Boks to glory in Japan four years ago.
"We were only leading 6-3 at halftime after having some dominance. They are a nuggety team and hats off to them. We had to grind the win out."
Nienaber said it wasn't rocket science how the Boks turned the screw in the second period.
"Rugby is not a complicated sport. If you get a good platform from the forwards, then you can create momentum that can create some space and then you have some backs who can make some magic," he said.
Captain Siya Kolisi admitted his team had been "a bit slow to get into the game and take opportunities".
"In the second half we took our opportunities from the set-piece and I am proud of the way the boys did that."
TMO INCONSISTENCIES
Scotland coach Gregor Townsend was left "really disappointed" by the result, which increases the ante in Pool B, which also includes world No 1 ranked team Ireland, Tonga and Romania.
"We were slow to get going, but we built into the game. There were opportunities, but not many in our attacking game.
"South Africa took possession and got some points and we never got the accuracy to really trouble them on the scoreboard.
"We only scored three points, that's unusual for us, credit to South Africa's defence."
Townsend also questioned the decision of the television match official not to examine a head-on-head tackle by Bok centre on Scotland No 8 Jack Dempsey just moments into the first half.
"There are still inconsistencies," the former Scotland and British and Irish Lions fly-half said, without wishing to further comment.
"I'm frustrated by that but more frustrated by our own performance."
Scotland skipper Jamie Ritchie was under no illusion of what lay ahead for the team.
"All the games are must-win now," he said, adding that the Ireland clash, at the Stade de France on October 7, "was always going to be a really tough game for us".
Ritchie added: "I'm really proud of how we fronted up physically... but we didn't execute under pressure.
"As a group we feel like we've got a lot to play for, we have three massive games ahead of us and the pressure is on us to perform.
"It's a setback in terms of we didn't get the result, but the good work we've done over the last few months hasn't gone."
