Former Aston Villa striker Marcus Allback poked home the winner after 73 minutes to leave United still with work to do to reach the knockout phase.
And Sir Alex Ferguson's side had only themselves to blame for a lacklustre approach and a host of missed chances.
John O'Shea and Wayne Rooney both wasted easy opportunities to score.
The scale of Copenhagen's win was emphasised by the fact that Allback's strike was their first goal in the Champions League proper and their first win - sparking wild celebrations at the final whistle.
United's mystifying attitude was typified by Ferguson's team selection, making six changes from the side that demolished Bolton in the Premiership on Saturday.
Copenhagen threatened after 23 minutes when Michael Silberbauer tested Edwin van der Sar low down at his near post.
And there was a scare for United four minutes later when Rooney went crashing painfully to the turf after a challenge by Brede Hangeland, who was booked by referee Wolfgang Stark.
But he recovered quickly to have United's best chance so far, a shot on the turn inside the area that rolled tamely wide.
Ferguson made a tactical switch of personnel at the interval, sending on Rio Ferdinand for Nemanja Vidic.
United were not at their best, but dominated after the interval and had opportunities to emphasise their superiority.
Ole Gunnar Solskjaer shot straight at Jesper Christiansen - although Copenhagen had a chance of their own when the unmarked Atiba Hutchinson headed wastefully off target.
O'Shea and Rooney were then both guilty of bad misses as United continued to let their limited Danish opponents off the hook.
Rooney set up O'Shea after 55 minutes, but he sidefooted wide with the goal at his mercy.
And Rooney was the culprit eight minutes later when he guided Gabriel Heinze's perfect cross wide from only eight yards.
Copenhagen were on the back foot, but when they were offered the chance to take advantage of United's generosity, they accepted the invitation gratefully to take the lead with 18 minutes left.
United failed to clear a corner, and when Hutchinson's header landed in the six-yard area, Allback poked home at the second attempt.
The growing frustration in United's ranks was shown when Cristiano Ronaldo was shown the yellow card for dissent.
United substitute Paul Scholes almost restored equality with six minutes left, but Christiansen raced quickly out of goal to block his attempted lob.
Ronaldo had the ball in the net with seconds left, but United's night was summed up when it was ruled out for offside.
Man Utd (from): Van der Sar, Kuszczak, Brown, Ferdinand, Vidic, Evra, Heinze, Silvestre, Lee, Ronaldo, Scholes, Carrick, O'Shea, Fletcher, Jones, Richardson, Rooney, Smith, Solskjaer.
Who are they?: A 14-year-old club that has won five domestic titles in that time. FC Copenhagen are the third Danish club to reach the Champions League group stages, after Aalborg and Brondby, and the first this century.
Goals: Allback 73.
Recent form: At present they top the Danish Superliga and have increased their lead from one point to three in the last fortnight with two wins and a draw. In five domestic matches in October they did not concede a goal, but are yet to score in the Champions League.
Man Utd: Van der Sar, Brown, Vidic (Ferdinand 46), Silvestre, Heinze (Evra 80), Fletcher (Scholes 71), Carrick, O'Shea, Ronaldo, Rooney, Solskjaer.Subs Not Used: Kuszczak, Smith, Richardson, David Jones.
European record: Bits and pieces in the Cup Winners' Cup and the Uefa Cup before this, their first time through to the Champions League group stages.
Booked: Ronaldo.
How did they get here?: They won the league last season and then saw off Finnish side MyPa before beating Ajax 2-0 away to go through to the group stages 3-2 on aggregate.
Att: 40,000
Ones to watch: Prior to this season only three of the Copenhagen squad had Champions League experience - former Chelsea midfielder Jesper Gronkjaer, Norwegian international defender Andre Bergdolmo and goalkeeper Jesper Christiansen. The stopper was on top form at Old Trafford and will need to be so again against Wayne Rooney and co.
Ref: Wolfgang Stark (Germany).
Did you know?: They have struggled on the pitch in the Champions League, but off it Copenhagen have recorded a win against Uefa who banned them from selling beer brewed by their sponsors in their first home match in the tournament. Club officials said that local custom "required" beer consumption during games and, having forced them to sell non-alcoholic beer against Benfica, Uefa has relented.
They say: "We've got a chance at home against all the teams. It'll be a good game against United and maybe Sir Alex can rest a couple of his star players for us!"Stale Solkbakken, FC Copenhagen coach
BIG MATCH STATS
Definitions of terms used:-
Champions League (CL) - only group phase matches and beyond of this competition which began in 1992-93. Champions Cup/Champions League - all matches played since it began in 1955-56 including qualification matches.European matches - all matches played in the major European tournaments (Fairs Cup, Uefa Cup, Cup Winners' Cup, Champions Cup, Champions League). Uefa Super Cups, Intertoto Cups and the old Intercontinental Cup competition are excluded.
Qualification
If Manchester United win this match, they are guaranteed a place in the last 16. A draw will also be enough if Benfica and Celtic also draw in the other match in this group.
Head-to-heads
Manchester United and FC Copenhagen met each other for the first time in the reverse fixture, United winning 3-0.
Copenhagen have hosted an English club before, losing 0-1 to Chelsea in the 1998-99 Cup Winners' Cup. The Red Devils have travelled to Denmark on one previous occasion, winning 2-6 at Brondby in 1998-99 - United's biggest away win in the Champions League and the only time that a club has scored six goals in an away match in the competition.
European history
Copenhagen are making their debut in the Champions League. They reached the group stage this season on their fifth attempt.
Manchester United have won the Champions Cup/Champions League twice (1968, 1999), the Cup Winners' Cup once (1991), the European Super Cup (1991) and the Intercontinental Club Cup (1999). They are taking part in their 12th Champions League campaign.
Current European form
FC Copenhagen have one point from their 0-0 draw with Benfica and share last place with the Portuguese club. They're yet to score a goal in the Champions League.
United lead Group F having won all three of their matches so far.
Player and disciplinary info
Andre Bergdolmo is FC Copenhagen's most experienced player in the Champions League with 35 appearances.
Gary Neville (94), Ryan Giggs (88) and Paul Scholes (87) are three players from the Manchester United squad who rank amongst the 10 players with most Champions League appearances.
Other miscellaneous facts
United manager Alex Ferguson has been in charge of 184 European matches in his career. His total is the second highest ever and he only trails Giovanni Trapattoni (196). Ferguson's Champions League total of 118 matches is a record for the competition. Ferguson will celebrate his 20th anniversary as Manchester United manager five days after this match.
United's Ole Gunnar Solskjaer and Copenhagen coach Stale Solbakken were both in the Norway squads for the 1998 World Cup and Euro 2000, and Copenhagen's Jesper Gronkjær scored against United when he was playing for Chelsea in the Premiership.
