"That's some of the best footy we've played in the middle of the field this season. Maybe we wanted to score too much."
It also appears to have come at a heavy cost with new No.8 Sean McMahon (ankle) sent for scans and facing a potentially lengthy spell on the sidelines.
McMahon hurt his ankle in a tackle just before the half-hour mark and tried to play on after having it strapped but didn't emerge after half-time.
"It would have to be something pretty significant for him not to come back out for the second half," Cheika said.
All of South Africa's points came via recalled five-eighth Morne Steyn, who kicked four penalties, two drop goals and expertly steered his team to victory.
Cheika spoke on match eve of persisting with the "Australian way" despite the noise around hoodoos, altitude and opposition tactics, and that's exactly what they did - just not for long enough.
They established a 10-3 lead after a Bernard Foley penalty and a debut Test try in the 14th minute to Scott Sio, who crossed under the posts to cap off an impressive multi-phase play peppered with smart offloads.
But the Springboks fought back and turned the match into a grind after Israel Folau's sin-binning six minutes from the break.
With only Folau to beat out wide, South African vice-captain Bryan Habana kicked ahead but was clipped as he tried to move past, prompting referee Wayne Barnes to produce his yellow card.
Cheika reckoned only a penalty was warranted but admitted Folau's absence cost them valuable momentum.
From there, the Springboks simply rode Steyn's boot to victory, with their set piece play particularly impressive in the second half as they won a string of scrum penalties.
Winger Reece Hodge missed three penalty attempts, two of them opportunities to put Australia ahead.
The other was an audacious early crack from approximately 64m that would have equalled the longest penalty scored in Test history.
It dropped less than a metre under the crossbar.
"That's the game, mate. We've got a very young, new side ... they'll learn from those experiences," Cheika said.