Australia looked out of sorts in the match, scoring a fortuitous try in the first half through Mark Gerrard before falling behind the Springboks on the scoreboard in the second half. With just minutes remaining the Wallabies scored a dramatic converted try to snatch the win.
"We struggled to get into the game with our attacking rhythm and we turned the ball over at times when on the attack in the first half," Gregan said after the match.
"They gave away some penalties when they were under pressure which we were able to convert into points in the first half, but we made some fundamental errors when we were building some pressure with the ball and that didn't allow us to play our attacking game."
However, Gregan was pleased with the gutsy effort shown by his players to dig deep and come away with the win, even though they weren't playing to the top of their form.
"I think the pleasing aspect, any time a team wins by a close margin, is that they were able to show some composure," the Wallabies skipper said.
"We showed a lot of character in grinding it out and doing it for each other."
Gregan, who overtook John Eales to become Australia's most capped captain in the match, indicated that one of the reasons for his team's below-par performance was a carry-over effect from last week's narrow 13-9 loss to the All Blacks in the Bledisloe Cup match in Brisbane.
"I think the boys were still flat on Wednesday, physically and emotionally," said Gregan.
"You kind of know when the team is up and we were certainly down at training on Wednesday."
"We refreshed and recharged as much as we could between Wednesday and Saturday but the boys hadn't gotten over the match against New Zealand."
"A fair bit went into the match in Brisbane but that is the nature of the Tri Nations and back-to-back Tests."
"It is always tough to get over a big match for the next one, regardless of the result, and the type of match we played last week was physically demanding, so it was a real mental challenge to get ourselves right."
The win keeps the Wallabies in the contest for the Tri Nations title with the next match against New Zealand in Auckland in two weeks time, a do-or-die encounter for Australia.