Denmark scored the first goal of the finals when Mariann Knudsen ran from halfway before slipping it past Kristin Hammarstrom for a magical strike.
But the hosts soon levelled as Sara Thunebro's free-kick was headed in Nilla Fischer, with what had previously been a Denmark-dominated match becoming a more even encounter.
After both sides had their chances, Sweden missed the first of their penalties as Lotta Schelin's weak spot-kick was saved by Stina Petersen.
The home side had another golden opportunity when Theresa Nielsen handled in the box: Kosovare Asllani stepped up this time, but again Petersen was the hero.
Sweden are awarded a penalty when Christina Ørntoft challenges Kosovare Asllani. takes the spot kick but is thwarted by Stina Petersen, who dives to her left to make the save. A golden opportunity for the hosts.
In the opening match in Halmstad, Finland held Italy to a 0-0 stalemate..
The Italians dominated for long spells but found themselves unable to break down a defensively-minded Finnish unit who failed to register a shot on target in the Group A encounter.
Goalkeeper Tinja-Riikka Korpela stood firm, first repelling two efforts from the surging Melania Gabbiadini, then denying midfielder Alessia Tuttino late on.
Earlier in the day, the Azzurre suffered a major blow when defender Elisabetta Tona was ruled out of the tournament with a leg injury suffered in training.
The result comes as a major disappointment for Italy, who find themselves in a tough Group A along with Sweden and Denmark.
The top two teams from each of the three groups reach the quarter-finals, plus the two third-placed teams with the best record.