With Austria beating Iceland 3-0 to top Group C, runners-up France will face the winners of Group D – likely to be England – in the last eight on Sunday.
After France full-back Eve Perisset was sent off, Ana-Maria Crnogorcevic’s fine header put Switzerland ahead early on.
But Camille Abily’s late free-kick saved France from a shock elimination.
England eye win to set up France meeting For France – ranked third in the Fifa rankings and one of the pre-tournament favourites alongside six-time winners Germany and in-form England – defeat in Breda would have ended their hopes of a first major tournament title.
Group winners Austria are making their tournament debut and are the lowest-ranked side in the group at 24th in the world.
Goals from Sarah Zadrazil, captain Nina Burger and substitute Stefanie Enzinger earned them a comfortable victory in Rotterdam over Iceland, who had already been eliminated.
France set for likely England tie
Switzerland knew they had to beat France to go through and a surprise win looked possible as they found themselves ahead against 10 players after 19 minutes.
Paris St-Germain’s Perisset, as the last defender, was shown a red card for bringing down Chelsea Ladies forward Ramona Bachmann.
From the resulting free-kick, Crnogorcevic’s powerful header from Martina Moser’s well-placed delivery gave Switzerland the lead.
However, with 14 minutes remaining, Abily’s free-kick was misjudged by Switzerland keeper Gaelle Thalmann, who had made several fine saves, as a relived France reached their third consecutive European Championship quarter-final.
England need only a point from their final group game against Portugal on Thursday to qualify as Group D winners and secure a quarter-final meeting with France in Deventer on Sunday.
France will be without captain and centre-back Wendie Renard after she received a second booking of the tournament.
England have not beaten France since 1974.
Austria await the runners-up in Group D. Scotland, currently bottom of the table, could qualify second but need to beat Spain by at least two goals and hope Portugal lose to England.