The home side scored all three goals before half-time as they secured second place in Group Four, with Northern Ireland second bottom with just four points.
The pick of the goals was scored in first-half injury-time by Andreas Laudrup, son of Swansea boss Michael.
Northern Ireland's evening got off to a disappointing start, Odense midfielder Emil Larsen putting the home side ahead with the first significant chance of the match with just eight minutes on the clock.
Accrington striker James Gray attempted to bring his side back into it fives minutes later only to see his shot blocked, while Matty Ball sent his free-kick wide moments later.
Denmark had attempts of their own but, one save from Ross Glendinning aside, were struggling to locate the target.
Josh Magennis gave them a helping hand when he conceded a 28th-minute penalty, allowing Denmark captain Mads Albaek to roll home his side's second.
The Aberdeen full-back had a chance to make amends with a free-kick of his own but could not beat the wall.
Simon Makienok had a three attempts to extend the home advantage but was unsuccessful and paid by being hauled off at half-time.
By that point it was already 3-0, though, Laudrup defeating Glendinning in added time with a wonderful finish.
With the result safe the hosts slowed down in the second period, while Northern Ireland picked themselves up and started enjoying more possession.
Neither goalkeeper was having to work too hard and a string of substitutions disrupted things further.
But it was Northern Ireland's replacements who looked most likely to get them a consolation.
Matt McClure tried his luck with 10 minutes to go and Billy Kee headed straight at the Frederik Ronnow from six yards after getting on the end of a floated cross by Seanan Clucas.