Full of confidence after getting the better of Norway, Wales had the game’s first chance, with Ethan Ampadu missing the target from distance inside the opening five minutes.
Denmark, meanwhile, grew into the first half as Rasmus Hojlund began to make his mark, bullying his marker before firing towards goal only for Danny Ward to get behind it.
Gustav Isaksen and Morten Hjulmand sent further chances off target, before Sorba Thomas fired a free-kick straight at Mads Hermansen down the other end during a moment of Welsh relief.
Isaksen then shot wide again prior to Oliver Provstgaard wasting the best chance of the first half, heading over from close range.
Denmark continued to dominate after the restart and Andreas Skov Olsen, who came on for the injured Pierre-Emile Hojbjerg after just 10 minutes, almost had an assist when Ward kept out Hojlund’s sliding attempt following the winger’s cross.
After withstanding their opponents' pressure, Wales almost scored themselves in fortuitous circumstances, with Thomas’ cross hitting Lewis Koumas’ back and forcing Hermansen into action.
As the goalkeepers began to take centre stage, Ward then produced a fantastic stop down the other end, tipping Isaksen’s header onto the post.
Nevertheless, Wales’ goalkeeper went from hero to villain shortly after.
A clumsy pass saw Ward give the ball away to Isaksen in the box, and he subsequently brought down the 25-year-old forward to give away a penalty.
Hojlund stepped up and, after a slow run-up, sent the home custodian the wrong way to give the visitors a precious lead.
With their previous six home games seeing both teams score, the Dragons were desperate to continue that trend and nick a late equaliser, but they ultimately fell to a fifth defeat in eight matches in all competitions (W1, D2), leaving them bottom of Group A4 with just two fixtures left.
In contrast, Denmark moved up to second and in strong contention for a quarter-final spot, although they will need to pick up points against someone other than Wales if they want to progress to the final eight.
Flashscore Man of the Match: Joachim Andersen (Denmark)