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Portugal started the game on the front foot, and had three golden opportunities to break the deadlock inside the opening 10 minutes.
Virtually straight from kickoff, Ivan Cavaleiro's deflected left-wing cross almost deceived Carlgren, only for the ball to drop inches over the bar.
Sweden struggled to clear the resulting corner and allowed the ball to drop to Ricardo Pereira on the edge of the area, who sent a fiercely-struck shot inches wide.
However, as the half wore on Sweden forced their way into the game, and they carved out a few chances of their own, most notably for John Guidetti.
Still it was Portugal who looked the most threatening, and they could have been in front at the break had Raphael Guerreiro's strike from range been on target.
The first chance of the second half fell to Sweden when Guidetti met Isaac Thelin's knockdown and lashed a half-volley just over the top of goal.
Substitute Toze's low effort nearly crept under the body of Carlgren on the hour mark, before fellow Portuguese replacement Iuri Medeiros cut inside from the right wing and drilled an effort past the upright.
Guidetti, the focus of much of his side's play, could have snatched a result for Sweden late on, but his effort from inside six yards was clawed away by Jose Sa to send the clash to extra time.
With both sides visibly tiring, chances were few and far between in extra time, and the game went to penalties. There was drama early on in the shootout as Toze's penalty struck the underside of the bar and only just crossed the line.
Carlgren then saved Ricardo Esgaio's penalty, but Abdul Khalili's tame spot kick allowed Portugal back in, only for Carlgren to keep out Carvalho's effort and ensure his country were crowned champions.