Michael Krohn-Dehli broke the deadlock for the Danes in the 24th minute, sending both Dutch captain Mark van Bommel and John Heitinga the wrong way and slotting his left-footed shot between goalkeeper Maarten Stekelenburg's legs.
The Oranje, who beat Morten Olsen's side in their first group match at the 2010 World Cup, looked for an equaliser, with Bayern Munich's Arjen Robben hitting the post in the 36th minute.
But the Danes stood firm and Bert van Marwijk's men were unable to find a way through in front of their 10,000-15,000 travelling fans in Kharkiv.
A second-half header from striker Mario Gomez was enough to see title contenders Germany through in Lviv, giving Joachim Loew's team their third straight win over Portugal in major tournaments.
Birthday boy Miroslav Klose had been warming up to replace Gomez, when he finally got the breakthrough, meeting Sami Khedira's cross to place his header outside the reach of the diving Patricio on 73 minutes.
European champions Spain on Sunday begin their bid to become the first team to win back-to-back titles when they take on Italy in Gdansk, northern Poland. in Group C at 1600 GMT. Ireland take on Croatia in Poznan in the other group match.
Euro 2012 kicked off in Warsaw on Friday, with Poland and Greece playing out an incident-packed 1-1 draw and Russia thrashing the Czech Republic 4-1 in the western city of Wroclaw.
Even without the start they wanted, Poland's noisy fans spent a lively night partying on the streets of Warsaw and the country's media assessed that the much-anticipated opening had gone without a hitch.
"We've passed our first test in Euro 2012," the Rzeczpospolita daily said.
But UEFA later announced disciplinary proceedings against the Russian football federation for "the improper conduct of its supporters" at the game, which saw fireworks set off and thrown and nationalist banners displayed.
One person with knowledge of the case told AFP that the Russian football federation were most likely to be fined and it was unlikely the country would be docked points or blocked from competing in future tournaments.
There was no direct mention of claims from a watchdog group that Czech player Theo Gebre Selassie was subjected to racial abuse by Russia fans, who were also suspected of beating up four volunteer stewards at the ground.
UEFA said only that they were probing the racial abuse claims -- which follow claims from Dutch players that some fans made monkey chants during their training session last week -- and was working with racism monitors to collect more evidence.
The incidents came amid lingering concerns about whether the two former eastern bloc nations were fully prepared to host European football's showpiece international competition.
Ukraine in particular has been at the centre of concerns over delays and cost-overruns in key infrastructure projects as well as sky-high accommodation costs and fears of racist violence.
Local media reported on Saturday that a Spanish football fan had been found murdered near host city Donetsk but police later said there was no connection to the tournament and the crime took place last month.
Ukraine will be hoping to overcome a focus on a boycott by some European countries over Kiev's alleged mistreatment of opposition leader Yulia Tymoshenko, who is serving a seven-year jail term for abuse of power and corruption.
She and her supporters claim the case against her is politically motivated.