Star midfielder William Carvalho saw his penalty saved by Patrik Carlgren, who made two stops in the all-important shootout.
Sergio Oliveira had hit the crossbar for Portugal in normal time, while John Guidetti was denied late on as neither side could find a goal as the game headed into an extra half an hour.
The inevitable shootout followed and Sweden, who lost 1-0 to England in the group stage, held their nerve to win the tournament for the first time.
Portugal came into the game as favourites after their 5-0 thumping of Germany in the semi-finals, and they settled into the match well, dominating the early spells of possession.
Ricardo Pereira went close inside the opening few minutes, drilling a low shot just wide of the upright, before Oliveira curled a superb 25-yard free kick around the wall, only to see it fly back off the crossbar.
Rui Jorge???s side continued to enjoy more time on the ball, but Sweden looked resilient and made it hard for them to be broken down ??? a long-range Raphael Guerreiro shot which went over being the only other meaningful attempt of the first half.
Guidetti grew into the game and managed to regularly make life difficult for the Portugal defence. He went close early after the break as he met Isaac Kiese Thelin???s knock-down with a rasping volley which flew just over the bar.
Moments later, Portugal skipper Oliveira went close again, firing wide from distance before being substituted in a surprise move by Jorge.
His side never looked the same after that change, although they went agonisingly close as substitute Iuri Medeiros cut in from the left before curling a shot just wide of the far post.
But Sweden looked more likely to win as the game approached extra time. Guidetti had a glorious chance to seal it with four minutes remaining as he charged through two players, only to see his placed effort across goal turned away by a fingertip save from Jose Sa.
In the added half an hour, Abdul Khalili went closest as he fired a curling effort from the left-hand edge of the area which flew just wide, but both sides looked tired and the game headed into a shootout.
In front of the Sweden fans, Guidetti started proceedings with a confident spot kick, before both sides added two apiece to leave it 3-2 to Hakan Ericson???s side.
Carlgren then made the first of his two saves, reading Ricardo Esgaio???s run-up and parrying away his effort from a comfortable height to his left, before Sa responded by keeping out Khalili???s tame effort.
Joao Mario and Victor Lindelof both found the net, leaving Carvalho with the unenviable task of having to put away Portugal???s fifth penalty.
But Carlgren guessed correctly as Carvalho went left, and the Sweden goalkeeper turned away his low strike to win the tournament for a jubilant Sweden.