England also beat Wales and Kenya.
Defending champions South Africa notched 104 points in victories over Samoa, Scotland and Russia in Pool B but Fiji topped the scoring charts with 112 points in wins against Argentina, Japan and Canada.
New Zealand topped Pool C but were stunned 14-12 by the U.S. for whom Perry Baker scored a dramatic late try that was converted after the buzzer by Madison Hughes.
The U.S. finished second to reach the quarter-finals.
"An extremely happy @ZTesty5 on @USARugby's stunning last second win over @nz7s to secure a Cup quarter final spot," captain Zack Test said on Twitter.
They will face South Africa on Saturday, when Fiji face Australia, New Zealand play Samoa and England take on Argentina.
The HSBC World Rugby Sevens Series will visit Cape Town, Wellington, Sydney, Las Vegas, Vancouver, Hong Kong, Singapore, Paris and London over the coming months.
It will showcase the sport as it prepares for its debut at the Olympics in Rio.
(Reporting by Martyn Herman, editing by Alan Baldwin)
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