In the end, they had to settle for six tries - scored by Matt Giteau (2), Jeremy Paul, Greg Holmes, Chris Latham and Mark Chisholm - while Stirling Mortlock added 16 points with the boot and Stephen Larkham knocked over an early drop goal.
"We're delighted with our performance but we know we can improve a great deal on it," said Connolly.
"We were loose in some areas. We know we left a lot of tries out there. And we can tighten up in certain areas but we'll talk about that."
The Australia scrum has come under intense pressure over the past 12 months and it looked like the massive South Africa pack was going to exploit that department in the early stages of the game.
But Connolly was happy with the way his forwards coped at that set-piece and felt the success in the scrum helped lay the foundations for their lethal backline.
"I think we had a couple of wonky ones (scrums) early on, but I think Guy Shepherdson did very well as did Holmes and Jeremy [Paul] in the middle," he said.
"I think the scrum improved and it's a great credit how guys like Shepherdson, who have had little experience, stepped up to the plate.
"They had a very experienced forward pack so we did very well."
Scrum-half George Gregan, the Australia skipper, echoed his coach's sentiments about the Wallaby pack, and admitted the win was a relief after their 32-12 loss to New Zealand in Christchurch last week.
"It was a good performance from the backs but that's been laid down by the awesome effort of our forwards," said Gregan.
"I thought it was outstanding. They really took on the challenge, rolled their sleeves up and got stuck in.
"We weren't happy with the way we performed in Christchurch and we realised there were a lot of areas we needed to improve and tighten up."
Gregan felt the victory was particularly sweet because the match was wrapped up at the halfway mark, when the score was 30-0.
"It's a rewarding win because we've had the chance to put them away in the past and we haven't," he added.
"Although we didn't score as many points in the second half, we didn't allow them to get back into the game and that's pleasing. It was a real shut-out.
"South Africa are a very proud rugby nation and they don't give in. They were competing right until the end."