The World Cup holders in the shortened version of the game had not been favoured in either final after some inconsistent performances on the world sevens circuits.
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.
Fiji, who won last week's Hong Kong sevens, were bidding for their first Commonwealth gold after taking silver behind New Zealand in the inaugural sevens in 1998 and again in 2002.
The look on her face said it all as she realised her mistake and that the match would go to extra-time.
New Zealand and England are currently playing their semifinal now at Robina Stadium.
South Africa took a 33-0 lead into half time, and by the time they were done they had scored a total of 8 tries.
But the Kiwis comfortably wound down the clock for victory and the gold medal.
"I got one in Delhi and was part of the team that lost in Glasgow (2014), so to get this win is incredible".
The men's Final followed and similarly New Zealand dominated the opening spell, going into the break 14-0.
In front of a packed stadium, Kelly Brazier broke two exhausted tackles and ran 75-metres to score the golden point victor.
New Zealand had opened up a 12-0 halftime lead after tries to Portia Woodman and Blyde, before Emilee Cherry exploited some poor tackling close to the line and Ellia Green outpaced the cover defence to send the contest into extra time.
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.
England's two rugby sevens teams battled back from bruising semi-final defeats against eventual winners New Zealand to leave the Gold Coast with Commonwealth bronze medals.
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