Silver medallists at last year’s Olympics, Sweden are the highest ranked team in the tournament, but failed to hit top gear in drawing with the Dutch and squeezing past the Swiss 2-1 in their opening two games.
Peter Gerhardsson’s women clicked into gear in Leigh and were 3-0 up by half-time.
Manchester City’s Filippa Angeldal struck twice before Carole Costa headed into her own net in first half stoppage time.
Kosovare Asllani’s penalty made it 4-0 before Arsenal striker Stina Blackstenius finally got on the scoresheet after having two earlier efforts ruled out.
Sweden will face whoever finishes second in Group D with Belgium, Iceland and Italy all in with a chance to finishing behind France in Monday’s matches.
The Netherlands lie in wait for the French in the last eight as they had enough to get through despite missing star striker Vivianne Miedema for the second consecutive game due to a positive case of coronavirus.
In front of a new record crowd of 22,596 for a women’s Euro group match not involving the hosts, Ana-Maria Crnogorcevic’s own goal put the Dutch in front in Sheffield.
Switzerland responded almost immediately when Geraldine Reuteler coolly finished from Romana Bachmann’s pull-back just three minutes later.
A draw would have been enough for Mark Parsons’ side to reach the knockout stages, but they made sure of the three points with three goals in the final six minutes
Substitute Romee Leuchter struck twice, either side of Victoria Pelova’s effort, which survived a lengthy VAR check for offside.