Buoyed by their 2-1 win over Denmark in Odense on the opening matchday in Group A4 on Friday, the side ranked fifth in the world made a bright start, taking the lead inside a quarter of an hour.
Neat play on the left side provided space for Matilda Vinberg to send a teasing cross into the box, forcing Welsh goalkeeper Olivia Clark to palm the ball into a dangerous area.
The ball fell to Filippa Angeldal, who applied the close-range finish as Clark failed to recover in time.
But any hopes Sweden had of that opening the floodgates failed to materialise, as they were frustrated by the hosts in a first half which did not see another shot on target.
Sweden piled on the pressure early in the second half, but once again could not turn their dominance into clear-cut chances against the team ranked 30th in the world.
Inside the final 20 minutes, Peter Gerhardsson's side came closest to a second when Rebecka Blomqvist sent Stina Blackstenius through, only for her effort to strike the post.
In front of the home crowd, Wales made the Blue-Yellow pay five minutes later when a ball into the box was handled by Emma Kullberg.
Kayleigh Barton stepped up for the Dragons and powered her penalty down the middle into the roof of the net, giving goalkeeper Jennifer Falk no chance.
Neither side were able to force a late winner, meaning Sweden move onto four points in the group after two matches.
Denmark's 3-1 win over Italy leaves the pair level on three, while Wales sit bottom of Group A4 with one point.
