The draw between Finland and Denmark means that both teams finish with only two points and England will now qualify if they get any sort of result against France in their final group match as at least a third best-placed side.
A draw against France would put England on the same record as Denmark - with two goals and a goal difference of -1 - but they would advance as they have a better UEFA coefficient.
Sweden exploded in the second half, netting three goals in the opening 12 minutes to preserve their unbeaten record and top Group A ahead of Italy who qualified as runners-up.
Finland became the first team to exit the tournament after a 1-1 draw with Denmark, who could still go through depending on results in the remaining games in the other groups.
Sweden finished on seven points with Italy on four ahead of Denmark and Finland on two apiece with the Finns heading home due to an inferior goal difference.
An own goal by Italy defender Rafaella Manieri set the Swedes on their way in the seaside town of Halmstad as she deflected substitute Therese Sjogran's pass into the net two minutes into the second period.
Two minutes later, striker Schelin got her third goal in three games after rounding the keeper and then set up Sweden's third goal just before the hour with a simple pull-back for Josfin Oqvist.
Italy's Melania Gabbiadini headed home unmarked to reduce the deficit 12 minutes from time.
"We didn't have many chances but the ones we got, we put away," Schelin told TV4 in a pitch-side interview afteer the Swedes had reached the knockout stage.
In Gothenburg, Denmark's Mia Brogaard struck a deflected 29th-minute goal that looked to have booked their place in the knockout stage but Finland equalised when Annica Sjolund rose to head home a corner three minutes from time.
Despite their meagre points tally, the Danes can qualify as one of the best third-placed teams depending on results in the final matches in Group B and C on Wednesday and Thursday.