United arrived in Denmark knowing another three points would confirm their qualification for the knockout stages, but a much-changed line-up from the side that thrashed Bolton at the weekend never really clicked and fell to a scrappy goal in the 73rd minute.
Sir Alex Ferguson decided to rest Rio Ferdinand and Paul Scholes, while Louis Saha, Gary Neville and Ryan Giggs did not make the trip after picking up slight knocks at the weekend.
And the decision to rest key personnel backfired spectacularly as United never found any sort of fluency and, even when they did threaten the Copenhagen goal, they were denied by a combination of inspired goalkeeping and poor finishing.
In a first half devoid of much goalmouth action, Wayne Rooney came closest to breaking the deadlock with a powerful shot straight at Jesper Christiansen from the edge of the area.
United sent on Ferdinand for Nemanja Vidic at half-time but it was the home side that grasped the initiative when Michael Gravgaard threatened with a header from a corner that Edwin Van der Sar claimed at his near post.
Cristiano Ronaldo wasted a glorious chance to play Ole Gunnar Solskjaer in on goal as United broke through the middle but the Portugal international over-hit a simple pass and Christiansen claimed bravely.
That let-off seemed to focus the home side and Atiba Hutchinson should have hit the target with a 12-yard header when left unmarked at the near post.
At the other end John O'Shea wasted the best United chance of the night as he arrived late in the box but contrived to steer a side-foot effort the wrong side of the upright.
Rooney was next to waste a clear-cut opportunity as he got the better of Gravgaard only to meet Gabriel Heinze's early cross with a poor header that flashed the wrong side of the upright.
But if United were wasteful in attack they were generous in defence, and when they failed to clear a corner effectively, Hutchinson turned the ball goalwards and Allback just about bundled it over the line to settle the match.
In the closing minutes United attempted to build up a head of steam but substitute Scholes was denied with a clever lobbed effort by Christiansen and, when Ronaldo rolled the ball home at the back post in stoppage time, the effort was correctly ruled out for offside.
To compound matters for United - who are by no means guaranteed to progress from Group F now - Solskjaer hobbled off with a hamstring injury right at the death.