Scott Curry, Ambrose Curtis, Tim Mikkelson and Sherwin Stowers scored in their pool decider with South Africa, who have been New Zealand's nemesis through much of the 2014-15 series.
South Africa had to work hard to earn their Olympic qualification, edging Samoa 19-12 and Kenya 15-5 after being down 5-0 early.
"It's a big moment for us," said South Africa's Frankie Horne. "We are very privileged to have made it this far. We have worked hard this season to qualify and it's an historical event for the game of sevens to be in the Olympics. We thoroughly deserve it but it's a great privilege and a great honour for us to be there and represent the game of sevens."
Australia won Pool A, despite a 17-all draw with England, after wins over France (38-7) and Russia (31-5). Australia are fifth, nine points adrift of England, and are fighting hard to leapfrog their old rivals in the next week.
SKY's Rugby Channel has live coverage from 9.30pm, while there is live streaming on worldrugby.org
New Zealand's pool results: New Zealand 24 South Africa 19, New Zealand 26 Samoa 19, New Zealand 22 Kenya 7