One of the pre-Euro 2012 favourites, the Dutch amassed 28 attempts to their opponents' eight at Metalist Stadium but it was a solitary 24th-minute strike by Brondby's Michael Krohn-Dehli that settled the opening Group B match.
Arjen Robben struck the post for the Netherlands in the 36th minute, while substitute Klaas-Jan Huntelaar forced a brave save from Danish goalkeeper Stephen Andersen towards the end of the second half.
Robin van Persie, meanwhile, looked like a different player to the one who cleaned up at England's end-of-season award ceremonies after scoring 30 Premier League goals for Arsenal in the finest season of his career to date.
The defeat left Holland facing an uphill climb to reach the quarter-finals, with games against Germany and Portugal still to come.
Van Marwijk refused to blame his players for their inability to beat Andersen, but he did criticise referee Damir Skomina for failing to award an 89th-minute penalty following an apparent handball by Lars Jacobsen.
"We had all those chances, but the ball didn't want to go in," he said.
"There was a lot of space. We put our substitutes on, we were more attacking, and we had more chances. When there's a penalty that doesn't go in your favour, it's unfortunate. But that's part of the game.
"We just have to beat Germany. It's not going to be easy, but that's the goal now and everyone knows that."
Van Persie's clumsiness typified his team's uncharacteristic lack of sharpness in front of goal, with one particular opportunity culminating in an air-shot that sent him tumbling to the turf.
Van Marwijk, though, felt that responsibility deserved to be shared between all of his attacking players.
"It wasn't only Robin van Persie," he said. "There were four, five or six players who had lots of chances. And the referee also had a chance to give us a penalty."
For all their opportunism, Denmark enjoyed just as much possession as Holland in the second half and collectively covered more ground (112,882 metres to 108,783) over the course of the match.
Van Marwijk, however, was keen to characterise the defeat as a smash-and-grab upset.
"In games like that, it's better to be defending," he said.
"You defend in small spaces and you expend less energy. I thought we could have done that a lot better."
Van Marwijk must now rally his troops for Wednesday's game with Germany, but he may take solace from the fact that Holland's 1988 Euro triumph -- their last major trophy -- also began in defeat, at the hands of the Soviet Union.
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