The Springboks celebrated their first Tri-Nations victory at Perth's Subiaco Oval since 1998 in front of a record crowd of 43,278.
It is also the first time in 23 years that the Wallabies have lost four straight Tests and they are now in danger of suffering their first whitewash in the Tri-Nations series when they tackle New Zealand in Auckland on September 3.
Speaking on national television, Gregan was also quick to deny suggestions he is heading to Japan once his Wallabies commitments are over.
"That's not true. We (his family) are very happy here in Australia and the more I travel the more I am happy to stay put," said the 32-year-old, Zambia-born scrum-half who needs two more Test caps to beat England prop forward Jason Leonard's world record of 114.
Despite this season's string of defeats, Gregan was upbeat when he was quizzed on Channel Seven's Sportsworld programme about the reasons for the loss.
"The team is always disappointed when we don't come up with a victory, but there was a lot of courage shown last night," Gregan said.
"The guys put a lot into that and we came up a little bit short, but the effort was good."
Australia were let down by poor handling, with South Africa's Bryan Habana twice breaking away to score after the Wallabies dropped the ball on the attack.
"You can't really explain it. There's no real formula for it," said Gregan, who admitted those errors cost his team dear.
"They're the things that turn these matches - a three-point differential.
"Twelve of those points came from us turning the ball over probably less than five metres from their (try) line.
"If you remove those two opportunities, those tries, it's a totally different ball game."
It was a similar story against the All Blacks in Sydney with winger Joe Rokocoko, like Habana, proving the big difference.
"It's the basics under pressure," added Gregan.
"If they hold up you'll get a good result. If they break down, as they have at crucial times in the past month, that's where the game turns."
Australia's hopes were also hit when fly-half Elton Flatley failed to take his place in the starting line-up after suffering dizziness during the warm-up.
"When it rains it pours," said Gregan. "We were ready to go. We did the gradual build-up but he had blurred vision. We tried to run another play but it got worse."
The setback leaves a big question mark over the playing future of the Queensland Reds' number 10 as earlier this season the 28-year-old Flatley sought specialist advice after suffering the latest in a serious of head knocks.
Meanwhile, experienced South Africa winger Breyton Paulse has been suspended for three weeks after being found guilty of kicking Australia prop Al Baxter.
The incident happened in the 77th minute, leaving Baxter with a cut to his face, and was not seen by match officials but he was subsequently charged by the Test's independent Citing Commissioner
The ban means Paulse, who has played more than 50 Tests for the Springboks, will miss next the showdown with New Zealand at Dunedin's 'House of Pain'.
The All Blacks must beat South Africa to have any hope of preventing them from retaining the Tri-Nations title.
At the hearing, Paulse agreed that he had kicked Baxter but claimed that he did so out of frustration at being held illegally by him and without any intention to inflict injury on the player.
He claimed that the contact made was in the shoulder/upper neck area, but Baxter gave evidence that the kick struck him just below his right eye.
The Judicial Committee, comprising Terry Willis and Michael Cashman from Australia, and Bruce Squire from New Zealand as chairman, viewed video footage of the incident, which showed it from a number of angles.
"The committee found that Paulse did not specifically target the Australian player's head, but that the kick was delivered towards the area of the head and thus carried an unacceptable risk of serious injury to the Australian player involved," Squire said in a statement.