Sweden opened well and took the lead after 14.53 when Lina Ljungblom put 1-0.
But the joy was short-lived and only 25 seconds later Denmark had equalized through Luleå's Nicoline Jensen after a Swedish defensive miss.
An unsuccessful pass in his own zone was then fought straight into the gap on Jensen.
The second period was only 24 seconds old when Denmark's Josefine Persson put 2-1.
Even that time, it was a Swedish miss who was behind the goal.
Josefin Bouveng then backed home and lost the puck at his own blue line.
Bouveng later took revenge when she thundered in 2-2 with a shot in powerplay.
Three Swedish defensive misses
When Denmark took the lead with 3-2, a third Swedish defensive miss came.
After a weak play, the Danes won the puck in the offensive zone and Josefine Persson scored her second goal of the day.
Denmark's Josefine Persson cheers after making 3-2 against Sweden.
Photo: TT
The away team looked for a while to go towards victory before Sweden managed a powerplay with about seven minutes left.
Josefine Holmgren then equalized to 3-3 with a powerful shot in numerical superiority.
The goal took the match to extra time where an early expulsion of Michelle Weis was punished for the Danes.
Michelle Löwenhielm put the decisive 4-3 on a free kick after one and a half minutes.