Celtic's travel sickness in the Champions League continued with an inept performance in Denmark.
Celtic's defence wilted under Copenhagen's aerial attacks
FC Copenhagen raced ahead within two minutes when Atiba Hutchinson shrugged off Neil Lennon's weak challenge to fire home from 12 yards.
Defensive deficiencies were exposed again after 27 minutes when Jesper Gronkjaer fired home from a corner.
Marcus Allback flicked in a third for Copenhagen before Jiri Jarosik grabbed a close-range consolation for Celtic.
The Bhoys, who were already assured of qualification, had only taken a point from 11 away fixtures in the Champions League and they never looked like claiming a win against the Danes.
Mark Wilson replaced Paul Telfer at right-back for Celtic after recovering from a broken foot sustained against Manchester United in September.
Free-kick specialist Shunsuke Nakamura was also relegated to the bench as manager Gordon Strachan started with Maciej Zurawski and Kenny Miller in attack with Jarosik in central midfield.
Celtic looked ill at ease from kick-off and fell behind within two minutes after Hutchinson shrugged off a weak challenge from Lennon to fire home the opening goal.
The Glasgow side failed to settle in the opening stages and looked to by-pass the midfield and play long balls to their front men.
The visitors could have fallen further behind after a quarter of an hour when they failed to clear a free-kick after Wilson fouled Gronkjaer.
Celtic captain Neil Lennon could not lead the side to an elusive away win
Marcus Allback's effort from six yards cannoned off the crossbar and to safety, but the home side looked likely to add to their lead.
Zurawski spurned Celtic's first chance on goal when he collected Wilson's pass and blasted over from 10 yards when Miller was unmarked.
The latter missed two glorious chances within a minute to restore parity.
Naylor's excellent cross found the unmarked Miller but he headed over.
The forward then raced through on goal from a Naylor through-ball, but his first touch round the keeper was poor and his subsequent shot was deflected for a corner.
Celtic were made to pay for their profligacy when Gronkjaer scored a tap-in after the SPL champions failed to clear a Linderoth corner.
Gronkjaer flashed a drive over the bar from 20 yards two minutes into the second half.
Allback compounded Celtic's misery after 57 minutes when the visitors melted under a free-kick played into the box and the Copenhagen forward flicked the ball home past a despairing Boruc.
Substitute Nakamura forced a save from Christiansen after 72 minutes in a rare attempt on goal by the visitors.
And Jarosik gave the travelling supporters something to cheer when he pulled a goal back for Celtic two minutes later, sliding home from six yards after an excellent Nakamura free-kick.
FC Copenhagen: Christiansen, Jacobsen (Norregaard 56), Hangeland, Gravgaard, Wendt, Silberbauer, Linderoth, Kvist, Gronkjaer (Berglund 89), Allback (Bergvold 81), Hutchinson. Subs Not Used: Gall.
Goals: Hutchinson 2, Gronkjaer 27, Allback 57.
Celtic: Boruc, Wilson, Balde, McManus (O'Dea 73), Naylor, Gravesen (Nakamura 68), Lennon, Jarosik, McGeady (Pearson 68), Zurawski, Miller. Subs Not Used: Marshall, Telfer, Riordan, Sno.
Booked: Miller, Jarosik, McManus.
Goals: Jarosik 75.
Referee: Bertrand Layec (France)