Emil Forsberg scored for Sweden in the first half and Zlatan Ibrahimovic made it 2-0 on a penalty kick at the start of the second.Denmark struggled to create any clear chances until Jakob Poulsen headed a corner toward the far post where Joergensen slotted home in the 80th minute.
"That goal could become a really, really important goal," Denmark coach Morten Olsen said.
The result means a 1-0 win in Copenhagen on Tuesday would see the Danes through.
Ibrahimovic, who scored eight goals in qualifying, downplayed the significance of the Danish goal.
"We won the game, that was the most important," he said.
Friends Arena in Solna was lit up in blue, white, and red to show solidarity with France following the terror attacks in Paris. Both teams wore black armbands and observed a minute of silence before the match.
Denmark looked stronger in the opening minutes but fell back as the Swedes surged toward the end of the first half. Marcus Berg missed two scoring chances and Ibrahimovic shot just wide before Sweden took the lead near halftime.
Breaking free on the right flank, Mikael Lustig sent a low cross into the area where an unmarked Forsberg scored with a direct hit inside the far post.
Denmark goalkeeper Kasper Schmeichel kept the score down before the interval with a magnificent fingertip save on Ibrahimovic's free kick.
Ibrahimovic made no mistake from the penalty spot, though, after Thomas Kahlenberg tripped Forsberg inside the area five minutes after the break. Schmeichel went for the right post but couldn't get there in time to stop the Sweden captain's powerful shot.
Ibrahimovic was substituted near the end of after signaling to the bench he wanted to come off. Sweden coach Erik Hamren said the striker was tired and out of breath.
Boosted by Joergensen's goal, Denmark pushed forward in the final minutes but couldn't find the equalizer.
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