Wales led 16-3 with just over a quarter of the match played at the Millennium Stadium but the World Champions hit back to run out 34-31 winners in front of a crowd of more than 60,000 in Cardiff.
Jones and Byrne now turn their attentions to a mouth-watering two-Test tour of New Zealand, with the Ospreys duo fully aware that Wales aren't far off where they'd like to be.
"We have to take ownership for all our errors both individually and as a team if we are going to improve," said Jones.
"We did really well for periods but we have to learn to show more composure and maturity to weather the storm when you don't have the ball.
"We didn't do that and we were beaten. That's what happens when you don't play well. We faced a very physical South Africa team who capitalised on our mistakes.
"Now we need to go down south (to NZ) and produce some big performances. Then we will know exactly where we are."
Those sentiments were echoed by Byrne, one of Wales' most consistent performers over recent seasons.
"We were firing on all cylinders at 16-3, but then we made a couple of errors, some by myself," added Byrne.
"In the end they had built too much of a lead for us to claw back. We made too many errors. We were 13 points up and we should have used our heads and just controlled the game.
"We didn't have to do anything flashy and just had to hit the corners.
"We were clinical in the opening period, but then errors started to creep in and we are just gutted by that."
Despite the disappointment of defeat to South Africa, Byrne is viewing the trip to New Zealand as a great chance to show exactly how far Wales have come under head coach Warren Gatland and his staff.
And while the odds may be heavily stacked in the All Blacks' favour, the 30-year-old knows victory is a realistic target - if Wales can reduce the error count in Dunedin and Hamilton.
"It doesn't get any easier, playing two Tests in New Zealand.
"But just like any team, we have to be confident going down there. This is Test rugby and everybody is up for it.
"We left it a bit too late for a Houdini-type win against South Africa. We cannot be doing that every time we play. We need to be comfortable in the last 10 or 15 minutes, having built a good lead.
"We just need to cut out the penalties and the errors. It's like a plague and just runs through a team when something goes wrong. It can be contagious and we cannot allow that to happen."