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.
A scoreless second half saw the New Zealand men secure their fifth Commonwealth Games gold medal.
The New Zealand women's side upset Australia 17-12 in their Commonwealth Games final.
The Australians, who had won the first Olympic gold medal in Rio in 2016 and dominated this season's women's sevens series, had been heavy favourites heading into the final between the trans-Tasman rivals.
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.
New Zealand Rugby CEO Steve Tew said it was a proud day on the Gold Coast.
The women got their campaign under way on Friday with wins over Kenya (45-0) and South Africa (41-0) before confirming their place at the top of their pool with a 24-7 win over Canada yesterday.
Fiji looked to recover when they were restored to a full complement but New Zealand captain 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.
"A number of our men's team were part of the silver medal winning team four years ago, so for them to experience gold today is a great result".
"I got one in Delhi and was part of the team that lost in Glasgow (2014), so to get this win is awesome". 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.
New Zealand took a 12-0 lead into the break, but a short restart earned Australia a penalty to begin the second half.
Australia came back from 12-0 down at halftime to lock the scores up at 12-12 in the closing stages with a conversion close to the posts.