Four-time gold medallists the United States and World Cup holders Spain are also in action in the last eight.
Canada's hopes of repeating their run to gold in Tokyo three years ago appeared to be over when they were docked six points by FIFA as a punishment after using a drone to spy on a New Zealand training session.
Coach Bev Priestman was suspended for a year.
But the Canadians beat Colombia 1-0 in Nice on Wednesday to make it three wins out of three, hours after an appeal to the Court of Arbitration for Sport against their points deduction was dismissed.
After a remarkable sequence of events, the eighth-ranked nation are through to the quarter-finals, and now face 2016 gold medallists Germany in Marseille on Saturday.
"We always knew that we could do it. The chances were stacked against us but we pulled through," defender Vanessa Gilles, the match-winner against France and Colombia, told Canada's CBC.
"We still believe the sanction was unfair, unjust and unprecedented."
Germany lost 4-1 to the USA in the group stage but went through with wins against Australia and Zambia, and Horst Hrubesch's team may be a step up on anything Canada have faced yet.
USA face Japan
The United States, targeting a record-extending fifth women's football gold, meet Japan in Paris in a repeat of the London 2012 final, which they won 2-1.
Under new coach Emma Hayes the USA won all three group games, with their attacking trio of Trinity Rodman, Mallory Swanson and Sophia Smith all impressing.
"Honestly we are kind of like sisters. This group we have right now is special and we are having a lot of fun," said Swanson.
The teams also met in the World Cup finals of 2011 and 2015, with one victory apiece, and Japan will be confident they can compete with the traditional powerhouse of the women's game.
Spain will take some stopping after the world champions eased through to the last eight with a 100 percent record in the group stage.
A side led by Ballon d'Or winner Aitana Bonmati will be strong favourites in Lyon against Colombia, who progressed as a best third-placed team.
However, Alexia Putellas warned her teammates not to take the South Americans lightly.
"It's going to be difficult to win every game," Putellas told FIFA.com, explaining that teams are setting up defensively against Spain.
"We've struggled to find our rhythm at times. Brazil, for example, really managed to stifle our attacks."
Marta ban
Brazil were handicapped against Spain -- a 2-0 defeat for the South Americans -- by the sending-off of captain Marta, one of the greats of the women's game.
Now suspended, the 38-year-old's chances of featuring again at her sixth Olympics hinge on Brazil beating France in Nantes.
Herve Renard's French team were not always convincing in the group stage but boast a lethal striker in Marie-Antoinette Katoto, the tournament's leading scorer with five goals.
"Marie carries the team because she is so clinical," said Renard after Wednesday's 2-1 win against New Zealand. "She doesn't need many chances to score."
France or Brazil will meet Spain or Colombia in the semi-finals, with the USA or Japan facing Canada or Germany.
