Carlgren threw himself to his right to save William Carvalho's spot kick and claim the crown for Sweden sparking joyous celebrations among the players and coaches.
Portugal looked the better side for much of the game. with Ricardo driving an early shot wide of the left-hand post and captain Sergio Oliveira crashing a free kick against the bar.
The Swedes battled their way into the game, creating a few half-chances of their own but often conceding possession for long periods to their more skilful opponents.
Undaunted, Sweden almost took the lead at the start of the second half, John Guidetti firing narrowly over the bar, but they were soon spending much of the game on the back foot again, with Portugal dictating the tempo until they tired late on.
Guidetti almost snatched a later winner for the Swedes but he missed a glorious chance and the final whistle went with the game still scoreless and the half-hour of extra time failed to produce a goal to separate the sides.
Each team converted three penalties before Carlgren saved from Toze. Portugal's Jose Sa, who had conceded one goal in five games in the tournament, then saved Abdullah Khalili's effort.
Both sides converted their next spot kicks before Carlgren saved Carvalho's penalty, triggering scenes of wild jubilation among the thousands of fans who had traveled to the Czech Republic to witness Sweden's first victory in a Euro final.
(Reporting by Philip O'Connor in Stockholm; editing by Ken Ferris)
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