Norway stunned the Danes by scoring from the first corner of the game in Norrkoping, Ingvild Stensland's ball into the box brushing off Marit Christensen and ending up in the back of the net after just three minutes.
But Denmark, who only made the knockout round at the expense of Russia after a drawing of lots with the two countries having finished the group stage level on two points, gradually gained the upper hand in a tight game.
Despite Denmark's burgeoning dominance, Norway countered quickly when given the chance and Ada Hegerberg shaved the top of the crossbar on the half-hour mark.
The Danes enjoyed long periods of possession in the second half but with the Norwegian defence funnelling most of their attacks infield, the Danes had difficulty turning that into chances.
Norway looked to be heading for the final until Denmark snatched an equaliser in the 87th minute, Mariann Knudsen heading in Pernille Harder's flick-on from a free kick.
That goal was enough to send the game to extra time but despite dominating throughout, Denmark failed to fashion a winner, which cost them dearly as the game went to penalties.
Goalkeeper Ingrid Hjelmseth was the hero for Norway, saving Denmark's first two spot kicks from Line Roddik and Theresa Nielsen. All four of Norway's efforts were successful, with Trine Ronning taking the final kick to put them into the final.
Germany beat hosts Sweden 1-0 in Gothenburg on Wednesday to book their place in the final, which will be played at the Friends Arena in Stockholm on Sunday.
(Reporting by Philip O'Connor in Norrkoping; Editing by Sonia Oxley)
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