Despite fashioning a slew of chances almost straight from the opening whistle, the tournament favourites were extremely poor in front of goal, and were punished for their profligacy by a fine finish from Michael Krohn-Dehli after 24 minutes.
The Dutch were similarly wasteful during the second half, their performance epitomised by the labouring Robin van Persie, who showed none of the killer instinct from his stellar 37-goal season with Arsenal.
The race for qualification from Group B is now wide open, after a stunning smash and grab result from Denmark, who defied the odds and now have a priceless three points to give their hopes of making the quarter-finals an undeniable boost.
With first choice left-back Erik Pieters unavailable, Bert van Marwijk began with Jetro Willems at the back who, at 18 years and 71 days old, became the youngest player ever to feature in the tournament.
It was the PSV youngster who had the first chance of the match, sending a crashing drive from long range over the bar with only three minutes gone.
Moments later, Van Persie spurned a more obvious opportunity when he belted a deflected cross off target from close range.
The Dutch appeared to be in control of proceedings, with Ibrahim Afellay sending another effort off target, before Arjen Robben’s dangerous centre was toe-poked into the grateful arms of Stephan Andersen.
However, for all their dominance, Netherlands lacked a cutting edge in front of goal and with 24 minutes gone, Denmark showed them how it should be done.
Krohn-Dehli still had work to do after picking up possession just outside the area, but the Brondby winger did not hesitate, nipping past his marker and drilling between the legs of Maarten Stekelenburg.
The goal sparked Denmark into life, who began to cause more problems by using their wide players to apply pressure down the flanks and pinging crosses into the heart of the Dutch penalty area.
They were almost pegged back after 36 minutes though, after a poor clearance from Andersen was pounced upon by Robben, who strode forward and bent an effort onto the base of the post.
Afellay skied another attempt following a marauding run into the box, but it was Van Persie’s miss a few minutes before half-time that left Van Marwijk shaking his head.
The Arsenal striker’s heavy first touch betrayed a glorious set up by Wesley Sneijder, but he still should have hit the target, instead of firing wide via a deflection by Andersen.
The interval did little to turn Van Persie’s fortunes, who wasted two further chances, with a trip over his own feet a particular cause of frustration, not least to the watching Klaas Jan-Huntelaar on the bench.
Mark Van Bommel called Andersen into action with a long range piledriver, before Afellay squeezed an attempt from the edge of the box inches wide of the post, as Netherlands seemed capable of everything but putting the ball into the back of the net.
Robben planted a header the wrong side of the post, but Holland could have been further behind on 70 minutes had Stekelenburg not gotten a touch onto a stinging drive by Krohn-Dehli.
The chances continues to come for the Dutch, with substitute Huntelaar wasting the pick of them when he failed to round Andersen after yet another incisive through ball from Sneijder.
But Denmark were holding firm at the back, and survived a late flicked header from Van Persie that inched over the bar. When a late penalty appeal for handball against Jacobsen went unanswered, the Dutch knew that their time was up.
Netherlands will have a difficult task getting their campaign back on track when they face fellow favourites Germany in their next group match, while Denmark square off with Portugal. Both fixtures take place next Wednesday.