The final quarter-final spot was decided when United States of America won its second match by beating Canada in the last match of the day.
Fiji was leading 14-5 in the first half with tries from Amenoni Nasilasila and Mesulame Kunavula.
The Olympic gold medallists will take on the US in the Cup quarter-finals while New Zealand play arch-rivals Australia.
Argentina had victories over Kenya, Wales and France and will go into their quarter-final as slight favourites over an English side that was thumped by South Africa 33-12.
Australia, New Zealand and Canada topped their pools Friday on the first day of women's competition at the Sydney Rugby Sevens tournament. As a team we have seen progression in such a short amount of time, although potentially not reflected in our finishing position.
The 2017 World Rugby player of the year criticised NZ Rugby, her employers, for finding excuses for not hosting a women's tournament when New Zealand's leg of the sevens world series takes place at Waikato Stadium in Hamilton on February 3-4.
This means that they have reached the final without conceding a single try.
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Defensive pressure was a key to Australia's semifinal win on Saturday.
Emma Tonegato scored in the same spot shortly before halftime to give Australia a 12-0 lead at the break.
Australia now faces its greatest test against New Zealand, which it hasn't beaten since the 2016 Olympic gold medal match in Rio de Janiero.
Australia and New Zealand have been battling it out for Sevens supremacy and they'll have another showdown in Sydney on Sunday afternoon.
"But they're a good side and going to be hurting from that defeat". "Here in Sydney we are focused on the basics because we know we can punish teams if we play to our game plan".
Woodman tops the try-scoring charts in the sevens world series with 22 - seven more than team-mate Michaela Blyde in second place.