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.
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 after 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.