Celtic had already qualified for the last 16 but their final game in Group F saw Gordon Strachan's side subside to another away defeat.
The Hoops are still to win away in the competition after 12 attempts and this fact left Strachan cold, ""Nothing can give us a lift and the disappointment overwhelms me," he said.
"We should have been celebrating with our fans because we are through to the last 16 but that defeat wiped it away," he added.
Goals by Atiba Hutchinson, Jesper Gronkjaer and Marcus Allback put Copenhagen into a comfortable lead by the 57th minute before Jiri Jarosik scored a consolation for the visitors.
Hutchinson played a one-two with Allback before slotting the ball past Artur Boruc from 12 yards.
Kenny Miller had two chances to equalise but Celtic fell further behind in the 27th minute when Gronkjaer volleyedhome.
Copenhagen made the game safe in the 57th minute when Allback beat Bobo Baldé and the hesitant Boruc in the penalty area to hook the ball in from eight yards out.
Copenhagen (4-4-2): Christiansen; Jacobsen (Norregaard, 56), Hangeland, Gravgaard, Wendt; Silberbauer, Linderoth, Kvist, Gronkjaer (Berglund, 89); Allback (Bergvold, 81), Hutchinson. Substitutes not used: Gall (gk).
Celtic (4-4-2): Boruc; Wilson, Baldé, McManus (O'Dea, 73), Naylor; Gravesen (Nakamura, 68), Lennon, Jarosik, McGeady (Pearson, 68); Zurawski, Miller. Substitutes not used: Marshall (gk), Telfer, Riordan, Sno.
Referee: B Layec (France).Reuse content