Bert van Marwijk's side created 28 chances at Metalist Stadium, hitting the post through Arjen Robben and having two penalty appeals turned down.
But only Krohn-Dehli was able to find the target and he did so in style, lashing a fierce angled drive through the legs of Maarten Stekelenburg in the 24th minute.
The result put the cat among the pigeons in the 'Group of Death', which also contains Germany and Portugal.
Jetro Willems became the youngest player at a European Championship, winning his third cap at the age of 18 years and 71 days, but showed no signs of being overawed by the occasion.
The PSV Eindhoven left-back blasted a long-range drive over Stephan Andersen's bar in the third minute as the Oranje signalled their attacking intent early on.
Robben's seventh-minute cross deflected kindly into the path of Robin van Persie - preferred up front to qualifying top scorer Klaas Jan Huntelaar - but the Arsenal captain sidefooted wide when he should have done better.
Barcelona midfielder Ibrahim Afellay sneaked down the left side of the area two minutes later and lashed an angled shot over the bar.
Robben was next to try his luck and managed to test Andersen, the Evian stopper getting down well to keep out the Bayern Munich forward's skidding left-footed attempt.
The Dutch looked totally in control, but were stunned in the 24th minute when Denmark took the lead against the run of play.
Simon Poulsen burst down the left and, after the Danes got lucky with his cross ricocheting straight to Krohn-Dehli, the Brondby winger despatched an angled left-footed drive through Stekelenburg's legs.
Netherlands looked rattled, but soon re-established their rhythm and pushed forward in search of the equaliser.
Ron Vlaar, playing at centre-back in place of the injured Joris Mathijsen, rose well to meet a Robben corner but could only direct his header wide, while Poulsen was fortunate not to concede a spot-kick when the ball hit his hand in the area.
Robben then latched on to a poor attempted pass from Andersen in the 36th minute and advanced to the edge of area before sending a left-footed shot against the post.
As half-time approached, though, Denmark's confidence grew and a well-worked move three minutes before the break almost put them two up, Stekelenburg diving to his left to save Krohn-Dehli's low effort.
Back at the other end, Van Persie should have levelled before the interval from Wesley Sneijder's pass but a poor first touch took him wide and allowed Andersen to narrow the angle and make the save.
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The Dutch continued in the same vein after the interval, but struggled to break down a well-organised Denmark defence.
Van Persie fell over his own feet when well placed and then had a right-footed shot on the slide kept out by Andersen.
Netherlands captain Mark van Bommel unleashed a vicious drive which brought another fine save out of Andersen, and John Heitinga powered a free header over the bar from Robben's corner.
The second half was following a similar pattern to the first, with Netherlands enjoying the lion's share of possession and chances, but Denmark remained dangerous on the break.
Poulsen's trickery down the left side of the box ended with a dangerous cross but nobody was on hand to convert, with Krohn-Deli fizzing in a low shot soon after which Stekelenburg did well to save.
In between those Denmark opportunities, Robben sent a free header wide from Afellay's cross, but the breakthrough was not forthcoming - prompting Van Marwijk to make a double substitution with 20 minutes left.
The Oranje boss introduced Huntelaar and Rafael van der Vaart for Afellay and Nigel de Jong, and the changes almost reaped instant rewards.
Schalke striker Huntelaar latched on to a stunning outside-of-the-boot pass from the influential Sneijder only to be denied by Andersen, with Van Persie's frustrating evening continuing when he fouled the keeper in attempting to pounce on the follow-up.
The Netherlands players were enraged in the 89th minute when referee Damir Skomina waved away penalty appeals after the ball struck Lars Jacobsen's hand and averted the danger from Huntelaar.
Van Persie flicked a header over the bar from Sneijder's cross two minutes into stoppage time, but it was not to be for Netherlands as Denmark held on for a famous win.