Carmona whistled a long shot just wide off the back post in the 14th minute for Spain’s closest call of the half.
Sweden didn’t produce a shot until the 42nd minute when Nathalie Bjorn looped a ball to the back post which Fridolina Rolfo volleyed, drawing a smart save from rookie goalkeeper Cata Coll.
Coll, just 22 and Barcelona’s reserve goalkeeper, made her debut for Spain in their round-of-16 win over Switzerland but has since kept her place in an extraordinary introduction to international football.
Putellas made way for Paralluelo shortly after the break which coincided with a momentum swing in favour of the Spanish.
Paralluelo’s tricky feet and speed immediately caused Sweden problems, stretching their defence.
Spain came close in the 69th minute when Alba Redondo fluffed Jenni Hermoso’s cross before Paralluelo gained possession for Redondo, who swept the ball into the side-netting while on the ground.
The breakthrough came when Sweden failed to deal with another Hermoso cross, with Paralluelo on the spot to volley home.
But Sweden weren’t done and equalised when Lina Hurtig knocked down Rolfo’s high ball to the unmarked Blomqvist, who volleyed smartly.
Then, in another twist, Spain’s Carmona received Teresa Abelleira’s corner at the top of the box and struck truly, sending Spain’s fans into delirium.
Until Tuesday night, Spain had never beaten Sweden and until this month they had never won a knockout match at the World Cup.
Spain had also encountered player mutiny and disharmony under Vilda’s leadership in the buildup to the tournament.
On Sunday, they have the chance to be crowned world champions.