The Canadians came into the tournament having lost all nine previous World Cup meetings with teams from Europe. On the plus side it tied the Netherlands 1-1 in its final round-robin game in Montreal, conceding the tying goal in the 87th minute.
Canada, which finished atop Group A, will face the Swiss, third in Group C, on Sunday at B.C. Place Stadium. The winner will meet No. 6 England or No. 11 Norway in the quarter-finals.
The Swiss (1-2-0) are ranked 19th in the world, compared to No. 8 for Canada (1-0-2).
The Canadian women have played Switzerland four times, with three victories and one draw. Their last meeting was a 2-0 Canada win at the 2013 Cyprus Cup.
It's Switzerland's first trip to the World Cup. Coach Martina Voss-Teckleburg's team finished third in Group C after losing 1-0 to Japan, thumping Ecuador 10-1 and being beaten 2-1 by Cameroon.
Cameroon's win — a two-goal comeback — over the Swiss was its first over a European opponent at the tournament.
The top four third-place finishers from the six groups advanced to the round of 16, along with the group winners and runners-up.
No. 7 Brazil faces No. 10 Australia in Moncton and No. 4 Japan meets the 12th-ranked Netherlands in Vancouver in the other round-of-16 knockout games on Canada's side of the draw. Norway plays England in Ottawa.
On the other — tougher — side of the draw, it's No. 16 China versus No. 53 Cameroon in Edmonton, the second-ranked U.S. versus No. 28 Colombia in Edmonton, No. 1 Germany versus No. 5 Sweden in Ottawa, and No. 3 France versus No. 18 South Korea in Montreal.
Round of 16 plays starts Saturday. The Dutch, Swiss and Cameroon advance to the knockout rounds in their first World Cup appearances.
Going home are Mexico, New Zealand and Nigeria, and newcomers Costa Rica, Ecuador, the Ivory Coast, Spain and Thailand.
The 2015 tournament, expanded to 24 teams from 16, marks only the second time that Canada has advanced out of the group stage in six tries.
The other time was in 2003 when Canada finished fourth after losing 2-1 to Sweden in the semifinals and 3-1 to the U.S. in the third-place game.
Canada's all-time tournament record is 5-11-5.
Follow @NeilMDavidson on Twitter