Neither side could break the deadlock for five minutes in the first half of extra-time, before New Zealand earned a penalty and Brazier ran nearly 80 metres to score the victor.
The BlitzBoks will face Fiji in the semifinals of the Sevens at the Commonwealth Games.
The women's side needed extra time to beat Australia and take the inaugural gold medal, with a heroic 80-metre effort from Kelly Brazier securing the win.
Brazier's dash had earlier silenced a boisterous local crowd at Robina Stadium, so they switched their support to Fiji for the men's decider.
Another turn of pace from Celestine Masinde doubled Kenya's lead to 14-0 at the break, before South Africa came roaring back with tries from Mathrin Simmers and Zenay Jordaan.
The Blitzboks lost 21-14 after leading 14-0 at halftime in their bronze medal play-off.
The women's final was also characterised by a strong first half performance from New Zealand and some desperate defence before Brazier stepped up to end home hopes of victory.
Fiji looked to recover when they were restored to a full complement but Scott Curry produced two desperate cover tackles to stop what would have been sensational tries and ensure they regained the title they lost in Glasgow. Two exceptional finals capped a superb tournament of firsts - the first women's competition and Jamaica and Zambia making their debuts in the men's event.
The New Zealand men's and women's sevens teams have claimed gold at the Gold Coast Commonwealth Games.
The final match of the day pitted Scotland against a South African side that had not conceded a try and the first five minutes saw the teams testing each other out with no quarter given before Specman went over in the corner after a quick tap.