The Scots were staring at a second heavy defeat in the EMC Test series after falling 21-3 behind in the 46th minute at Murrayfield following a second try for Adriaan Strauss.
But replacement scrum-half Henry Pyrgos crossed in the 51st minute following a well-worked line-out in what was then only the second attack of note Scotland had produced – the first had also come from a penalty kick into touch.
The try prompted a complete transformation in the momentum of the game with Scotland putting the tourists under relentless pressure for the remaining half-hour.
But some desperate defending, crucial calls from Irish referee George Clancy and poor decision-making meant the score stayed at 21-10.
"The last 30 minutes it was either a case of lie down and die or stand up and fight, and I thought we stood up and fought," Brown said.
"As a captain I want a side that is going to stand up and fight, but next week we need to do that from the first whistle.
"We need to learn our lessons and next week it's about really getting fired into Tonga."
Scotland had numerous chances in the last half hour, with lock Jim Hamilton coming closest when he went over the line with the ball in hand before he was pushed back.
Clancy made marginal calls on line-out and scrum offences that relieved the pressure, but Scotland were also guilty of wasting opportunities, notably when Greig Laidlaw and Ruaridh Jackson failed to execute short-range kicks over the top of the Springboks defence, and when Tim Visser knocked on Jackson's close-range pass before Nick De Luca went over in the corner.
But chances had been few and far between in the first half with Scotland camped in their own half. South Africa were pulled back for obstruction as they were about to drive over for an opening try, but soon worked another unstoppable maul for Strauss to cross midway through the first half.
The hooker was handed a simple chance to run in for his second try when he intercepted Mike Blair's pass.
Brown agreed with head coach Andy Robinson that conceding too many penalties was the root cause of Scotland's first-half troubles.
"We needed to be very disciplined," Brown said. "And I felt in the first half we weren't and our energy was a little bit down."
Lock Richie Gray suffered concussion during the first half on Saturday and is a doubt for the final match of the EMC Test series against Tonga in Aberdeen next Saturday.