Wales beat Fiji 19-14 to ensure they finished the group stage unbeaten, while New Zealand, who despite dominating the annual sevens series have not been world champions since 2001, also won all three of their matches.
They were given a scare by the United States in their final pool match, however, having to come back from 19-5 down to eventually prevail 26-19, with a late try to David Raikuna ensuring they took all three points.
"We made a few mistakes and they pounced on them, and they made us pay for it," captain Tim Mikkelson said.
"They are a good team and we showed a lot of fight to come back."
South Africa did not concede a point in beating Scotland, Japan and Russia and will play Fiji, who along with France were one of the two best-placed group runners-up to progress to the quarter-finals.
France will take on Kenya, who beat Samoa 17-12 thanks to a late try to Michael Wanjala in their final pool match to decide who would top their group.
The final quarter-final will be between England and Australia.
England topped their group with wins over Portugal, Hong Kong and Argentina, while Australia also finished in first place despite drawing their final pool match with France 17-17, having held a 17-0 lead at one stage.
Terry Bouhraoua kicked a penalty drop-goal with less than two minutes remaining to give his side a share of the spoils.
"A bit disappointed to have the draw, we switched off a little bit in the second half," Australia's Luke Morahan said.
"We played pretty well in the first two games but just let ourselves down a little bit at the end. It is good to get through."
Arguably the biggest upsets were in Pool F, where Portugal beat Argentina 17-10 but then went down 17-14 to Hong Kong, having led 14-5 with two minutes remaining.
Rugby sevens has been introduced into the Olympic Games and will make its debut at Rio de Janeiro in 2016.
- Sports & Recreation
- David Raikuna
- Tim Mikkelson