Wales ended a 14-match losing streak to Fiji by upsetting the Olympic rugby champion 28-15 in the Sydney Sevens on Saturday.
A hat trick of tries by Morgan Williams led Wales to its first win over Fiji since 2013, and ultimately to first qualifier in Pool B.
England also topped its pool after beating world series leader South Africa 21-15, handing the Blitzboks their first defeat since they lost to England in December in the Cape Town final.
South Africa led 15-7 with four minutes to go, but Dan Norton crossed for a converted try and James Rodwell scored the match-winner from a crosskick by Ollie Lindsay-Hague.
Beside England, New Zealand was the only other side to win all of its pool matches. The United States was the other pool winner, beating Argentina and Canada but losing to Russia 17-14 for the first time in five years.
In the quarterfinals on Sunday, it's: England vs. Argentina, New Zealand vs. Fiji for the second straight week, United States vs. South Africa, and Wales vs. Australia.
Wales finished the pool round by drawing with Samoa 7-7, a team the Fijians crushed 40-0.
Before England overcame South Africa, it also had to come back to beat Kenya 19-12, and defeated Japan 26-0.
New Zealand beat Australia 26-14 in a rematch of last year's final, rolled over Papua New Guinea, and came from behind to defeat Scotland 14-7.
Argentina advanced as second qualifier behind the United States after drawing with Canada 17-17. Pat Kay scored a last-seconds try for Canada but couldn't convert it.
Meanwhile, in the women's tournament, Canada won the final, beating the United States 21-17 with a late try by Bianca Farella.
Naya Tapper scored three tries for the U.S. but Farella had the last say.
The all-North American final was set up by two big upsets in the semifinals: Canada silenced the home fans by topping Australia 12-7, and the U.S. downed New Zealand 19-12. New Zealand beat Australia 19-0 for third place.