The Olympic silver medallists were strong favourites to progress against a Belgian side playing in their first ever knockout game at a major women's tournament.
However, the number two ranked side in the world were frustrated by Belgium goalkeeper Nicky Evrard, poor finishing and a controversial VAR review for 92 minutes.
Sweden's preparations had been hit by a trio of positive coronavirus cases for Jonna Andersson, Hanna Glas and Emma Kullberg.
But they still dominated from start to finish with 34 attempts to Belgium's three.
Filippa Angeldal forced Evrard into the first of a series of saves inside the first 10 minutes.
The Manchester City midfielder, who scored twice in Sweden's 5-0 win over Portugal in their final group game, then fired over with a clearer opening from the edge of the box.
Sweden thought they had made the breakthrough on 26 minutes when Stina Blackstenius coolly slotted past Evrard from Kosovare Asllani's through ball.
However, for the third time in the tournament, the Arsenal striker was denied by a VAR review for offside which harshly ruled the goal out.
The fluency of Sweden's attack disappeared after the break as they struggled to break through the mass ranks of Belgian defence.
Set-pieces then became the biggest threat from Peter Gerhardsson's side.
Blackstenius should have scored when she headed too close to the excellent Evrard.
The Gent goalkeeper was even equal to Nathalie Bjorn's effort in the final minute of stoppage time, but the rebound fell kindly for Sembrant to smash the rebound home and spark wild scenes of celebration among the Swedish squad.