The Dutch fell to defeat despite having over 60 percent of possession and 32 shots at goal.
It was Sparta Rotterdam midfielder Krohn-Delhi who handed Morten Olsen's side the three points from the Group B match when he drove the ball through the legs of goalkeeper Maarten Stekelenburg on 24 minutes.
Denmark next face Portugal while the Netherlands must avoid defeat against bitter foes Germany later on Wednesday night to avoid tumbling out of the tournament at the group stage two years after they lost to Spain in the World Cup final.
The Dutch went on to win Euro 88 after losing their opening match to the old Soviet Union, but they will need to call on all their powers of recovery with too many of their leading performers suffering form lapses in the key moments of this contest.
Despite being clearly superior on the ball to Denmark, men such as Arjen Robben, Ibrahim Afellay and Robin van Persie could not unearth the goals their country needed to avoid defeat on a hot and humid night in Ukraine.
With Simon Kjaer and Daniel Agger in supreme form at the heart of the Danish defence, the Dutch eventually ran out of ideas which will be the most worrying aspect of the defeat for coach Bert van Marwijk overseeing a country that has been littered with episodes of self-implosion over the years at such tournaments.
The Netherlands have made it out of the group stages in their last six appearances at the European Championship. They are the only side to have done this, but may need to beat Germany to salvage that record.
Having scored 37 goals in qualifying - more goals than any other country - the Netherlands were expected to wallow in goals against a country who had not beaten them in open play since 1967.
But the night became a struggle before a stadium that was disappointingly not full as Van Persie slipped a shot wide of goal on seven minutes before Lars Jacobsen dashed back to make a vital intervention as Robben tried to play in Wesley Sneijder rather than opt to shoot.
Krohn-Delhi had no such issues as he bounded beyond Johnny Heitinga before finishing with some expertise.
The chances came and went for the Dutch from that point onwards with Robben hitting a post after an error by Danish goalkeeper Stephan Andersen, who later dived to nudge a shot by Mark van Bommel past a post.
He also saved from substitute Klaas-Jan Huntelaar as the Dutch finished with Huntelaar and Van Persie on the pitch.
It became increasingly silly to watch with the Netherlands dominant but toothless. Having said that, Denmark were also rescued by two moments of good fortune as Simon Poulsen and Lars Jacobsen avoided penalties in either half despite clearly handling the ball on both occasions.
Sneijder created 10 goalscoring chances for the Netherlands. No player has produced more in a European finals match in the past 32 years, but it meant little as he slumped to his knees at full-time.
Denmark had probably worked hard for their slices of good fortune.